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Properties

Label 24-1400e12-1.1-c1e12-0-2
Degree $24$
Conductor $5.669\times 10^{37}$
Sign $1$
Analytic cond. $3.80944\times 10^{12}$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·9-s + 12·11-s + 12·19-s − 48·29-s + 6·31-s + 12·41-s − 6·59-s − 18·61-s + 36·71-s − 6·79-s + 45·81-s + 12·89-s − 108·99-s − 78·101-s + 12·109-s + 108·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 108·171-s + 173-s + ⋯
L(s)  = 1  − 3·9-s + 3.61·11-s + 2.75·19-s − 8.91·29-s + 1.07·31-s + 1.87·41-s − 0.781·59-s − 2.30·61-s + 4.27·71-s − 0.675·79-s + 5·81-s + 1.27·89-s − 10.8·99-s − 7.76·101-s + 1.14·109-s + 9.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8.25·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{36} \cdot 5^{24} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(3.80944\times 10^{12}\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{36} \cdot 5^{24} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.42320874\)
\(L(\frac12)\) \(\approx\) \(10.42320874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + 683 T^{6} + p^{6} T^{12} \)
good3 \( 1 + p^{2} T^{2} + 4 p^{2} T^{4} + 11 p^{2} T^{6} + p^{5} T^{8} + 4 p^{4} T^{10} + p^{4} T^{12} + 4 p^{6} T^{14} + p^{9} T^{16} + 11 p^{8} T^{18} + 4 p^{10} T^{20} + p^{12} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 6 T + 10 T^{3} + 222 T^{4} - 42 T^{5} - 3181 T^{6} - 42 p T^{7} + 222 p^{2} T^{8} + 10 p^{3} T^{9} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 4033 T^{6} + p^{6} T^{12} )^{2} \)
17 \( 1 + 36 T^{2} + 996 T^{4} + 13086 T^{6} + 44448 T^{8} - 3578544 T^{10} - 84671629 T^{12} - 3578544 p^{2} T^{14} + 44448 p^{4} T^{16} + 13086 p^{6} T^{18} + 996 p^{8} T^{20} + 36 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 - 6 T - 12 T^{2} + 134 T^{3} + 126 T^{4} - 1350 T^{5} + 1467 T^{6} - 1350 p T^{7} + 126 p^{2} T^{8} + 134 p^{3} T^{9} - 12 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 93 T^{2} + 4632 T^{4} + 152715 T^{6} + 3741027 T^{8} + 74385876 T^{10} + 1535077577 T^{12} + 74385876 p^{2} T^{14} + 3741027 p^{4} T^{16} + 152715 p^{6} T^{18} + 4632 p^{8} T^{20} + 93 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 + 12 T + 126 T^{2} + 715 T^{3} + 126 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
31 \( ( 1 - 3 T - 48 T^{2} + 163 T^{3} + 981 T^{4} - 1800 T^{5} - 21897 T^{6} - 1800 p T^{7} + 981 p^{2} T^{8} + 163 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( 1 + 132 T^{2} + 8868 T^{4} + 388526 T^{6} + 12594816 T^{8} + 324538704 T^{10} + 9478847859 T^{12} + 324538704 p^{2} T^{14} + 12594816 p^{4} T^{16} + 388526 p^{6} T^{18} + 8868 p^{8} T^{20} + 132 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 - 3 T + 105 T^{2} - 189 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
43 \( ( 1 - 153 T^{2} + 13041 T^{4} - 682705 T^{6} + 13041 p^{2} T^{8} - 153 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( 1 + 132 T^{2} + 8772 T^{4} + 289446 T^{6} - 527952 T^{8} - 713095392 T^{10} - 47085110341 T^{12} - 713095392 p^{2} T^{14} - 527952 p^{4} T^{16} + 289446 p^{6} T^{18} + 8772 p^{8} T^{20} + 132 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 - 60 T^{2} - 564 T^{4} - 82418 T^{6} + 619560 T^{8} + 581599656 T^{10} - 25774436685 T^{12} + 581599656 p^{2} T^{14} + 619560 p^{4} T^{16} - 82418 p^{6} T^{18} - 564 p^{8} T^{20} - 60 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 3 T - 144 T^{2} - 143 T^{3} + 13479 T^{4} + 2334 T^{5} - 911101 T^{6} + 2334 p T^{7} + 13479 p^{2} T^{8} - 143 p^{3} T^{9} - 144 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 9 T - 36 T^{2} - 1217 T^{3} - 3663 T^{4} + 37818 T^{5} + 605757 T^{6} + 37818 p T^{7} - 3663 p^{2} T^{8} - 1217 p^{3} T^{9} - 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 138 T^{2} + 4845 T^{4} + 111118 T^{6} + 10336554 T^{8} - 1688876910 T^{10} - 253781775291 T^{12} - 1688876910 p^{2} T^{14} + 10336554 p^{4} T^{16} + 111118 p^{6} T^{18} + 4845 p^{8} T^{20} + 138 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 9 T + 132 T^{2} - 765 T^{3} + 132 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
73 \( 1 + 294 T^{2} + 43941 T^{4} + 4869554 T^{6} + 458644698 T^{8} + 37183470174 T^{10} + 2756049047421 T^{12} + 37183470174 p^{2} T^{14} + 458644698 p^{4} T^{16} + 4869554 p^{6} T^{18} + 43941 p^{8} T^{20} + 294 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 + 3 T - 228 T^{2} - 231 T^{3} + 36033 T^{4} + 228 p T^{5} - 3285601 T^{6} + 228 p^{2} T^{7} + 36033 p^{2} T^{8} - 231 p^{3} T^{9} - 228 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 384 T^{2} + 68736 T^{4} - 7236429 T^{6} + 68736 p^{2} T^{8} - 384 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 6 T - 114 T^{2} - 162 T^{3} + 6936 T^{4} + 57324 T^{5} - 741557 T^{6} + 57324 p T^{7} + 6936 p^{2} T^{8} - 162 p^{3} T^{9} - 114 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( ( 1 - 273 T^{2} + 45213 T^{4} - 5241265 T^{6} + 45213 p^{2} T^{8} - 273 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.07336335661647007339600864297, −2.89855879607238860123546997374, −2.88461905711385455488082261572, −2.72411470113670394356475524805, −2.65387271910131043190750662343, −2.58142770193775324446693150148, −2.46256420873902269558327889715, −2.45559456837015036332936229284, −2.13037481172763619799067264912, −2.05294492260353163343571560232, −2.01108845809719222690442437510, −1.84671688051961534281591628502, −1.82974252095766487049261586765, −1.73415244966530852164689536454, −1.61135379256950364304935676904, −1.39606468497451227450743790008, −1.39177921570861694200418669315, −1.37152031369674969522702897856, −1.21024445792985370770897925316, −1.05883929694328166404571344149, −0.57043009776017601677176982284, −0.55088525838216558279183907065, −0.52165761968286040169901395688, −0.46411040280448948931566954867, −0.24387707304330304157100125803, 0.24387707304330304157100125803, 0.46411040280448948931566954867, 0.52165761968286040169901395688, 0.55088525838216558279183907065, 0.57043009776017601677176982284, 1.05883929694328166404571344149, 1.21024445792985370770897925316, 1.37152031369674969522702897856, 1.39177921570861694200418669315, 1.39606468497451227450743790008, 1.61135379256950364304935676904, 1.73415244966530852164689536454, 1.82974252095766487049261586765, 1.84671688051961534281591628502, 2.01108845809719222690442437510, 2.05294492260353163343571560232, 2.13037481172763619799067264912, 2.45559456837015036332936229284, 2.46256420873902269558327889715, 2.58142770193775324446693150148, 2.65387271910131043190750662343, 2.72411470113670394356475524805, 2.88461905711385455488082261572, 2.89855879607238860123546997374, 3.07336335661647007339600864297

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.