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Properties

Label 10-3344e5-1.1-c1e5-0-0
Degree $10$
Conductor $4.181\times 10^{17}$
Sign $1$
Analytic cond. $1.35742\times 10^{7}$
Root an. cond. $5.16739$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7·5-s − 8·7-s − 3·9-s + 5·11-s + 4·13-s − 7·15-s + 10·17-s − 5·19-s + 8·21-s − 5·23-s + 15·25-s + 2·27-s + 16·29-s − 31-s − 5·33-s − 56·35-s − 37-s − 4·39-s + 28·41-s − 4·43-s − 21·45-s + 4·47-s + 14·49-s − 10·51-s + 10·53-s + 35·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 3.13·5-s − 3.02·7-s − 9-s + 1.50·11-s + 1.10·13-s − 1.80·15-s + 2.42·17-s − 1.14·19-s + 1.74·21-s − 1.04·23-s + 3·25-s + 0.384·27-s + 2.97·29-s − 0.179·31-s − 0.870·33-s − 9.46·35-s − 0.164·37-s − 0.640·39-s + 4.37·41-s − 0.609·43-s − 3.13·45-s + 0.583·47-s + 2·49-s − 1.40·51-s + 1.37·53-s + 4.71·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 11^{5} \cdot 19^{5}\)
Sign: $1$
Analytic conductor: \(1.35742\times 10^{7}\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 11^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.735498320\)
\(L(\frac12)\) \(\approx\) \(8.735498320\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
11$C_1$ \( ( 1 - T )^{5} \)
19$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + T + 4 T^{2} + 5 T^{3} + 14 T^{4} + p T^{5} + 14 p T^{6} + 5 p^{2} T^{7} + 4 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - 7 T + 34 T^{2} - 117 T^{3} + 68 p T^{4} - 813 T^{5} + 68 p^{2} T^{6} - 117 p^{2} T^{7} + 34 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 8 T + 50 T^{2} + 220 T^{3} + 790 T^{4} + 2294 T^{5} + 790 p T^{6} + 220 p^{2} T^{7} + 50 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 36 T^{2} - 160 T^{3} + 870 T^{4} - 2512 T^{5} + 870 p T^{6} - 160 p^{2} T^{7} + 36 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{5} \)
23$C_2 \wr S_5$ \( 1 + 5 T + 83 T^{2} + 440 T^{3} + 3258 T^{4} + 14806 T^{5} + 3258 p T^{6} + 440 p^{2} T^{7} + 83 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 16 T + 182 T^{2} - 1380 T^{3} + 9268 T^{4} - 50904 T^{5} + 9268 p T^{6} - 1380 p^{2} T^{7} + 182 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + T - 6 T^{2} - 201 T^{3} + 336 T^{4} + 4615 T^{5} + 336 p T^{6} - 201 p^{2} T^{7} - 6 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + T + 145 T^{2} + 68 T^{3} + 9330 T^{4} + 2278 T^{5} + 9330 p T^{6} + 68 p^{2} T^{7} + 145 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 28 T + 436 T^{2} - 4716 T^{3} + 40138 T^{4} - 280428 T^{5} + 40138 p T^{6} - 4716 p^{2} T^{7} + 436 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 4 T + 162 T^{2} + 272 T^{3} + 10618 T^{4} + 7726 T^{5} + 10618 p T^{6} + 272 p^{2} T^{7} + 162 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 4 T + 119 T^{2} - 576 T^{3} + 9462 T^{4} - 32760 T^{5} + 9462 p T^{6} - 576 p^{2} T^{7} + 119 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{5} \)
59$C_2 \wr S_5$ \( 1 - T + 211 T^{2} - 336 T^{3} + 21526 T^{4} - 28830 T^{5} + 21526 p T^{6} - 336 p^{2} T^{7} + 211 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 16 T + 113 T^{2} + 240 T^{3} + 1418 T^{4} + 18176 T^{5} + 1418 p T^{6} + 240 p^{2} T^{7} + 113 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 9 T + 238 T^{2} - 2279 T^{3} + 26376 T^{4} - 225567 T^{5} + 26376 p T^{6} - 2279 p^{2} T^{7} + 238 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 7 T - 8 T^{2} - 617 T^{3} + 4226 T^{4} + 64625 T^{5} + 4226 p T^{6} - 617 p^{2} T^{7} - 8 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 8 T + 225 T^{2} - 1360 T^{3} + 22806 T^{4} - 114320 T^{5} + 22806 p T^{6} - 1360 p^{2} T^{7} + 225 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 6 T + 367 T^{2} + 1808 T^{3} + 55950 T^{4} + 210804 T^{5} + 55950 p T^{6} + 1808 p^{2} T^{7} + 367 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 4 T + 186 T^{2} + 448 T^{3} + 21890 T^{4} + 36478 T^{5} + 21890 p T^{6} + 448 p^{2} T^{7} + 186 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 31 T + 677 T^{2} - 9948 T^{3} + 122978 T^{4} - 1227418 T^{5} + 122978 p T^{6} - 9948 p^{2} T^{7} + 677 p^{3} T^{8} - 31 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 13 T + 205 T^{2} - 1008 T^{3} + 16658 T^{4} - 71094 T^{5} + 16658 p T^{6} - 1008 p^{2} T^{7} + 205 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.08832947017555354882269059197, −5.06075910948859889058815137531, −5.02979934964389408215927178843, −4.56517972860610334833553134219, −4.53619344600921363840534939668, −4.14194120348923618317526728211, −4.06672036594375677088553941964, −3.91104710240959013538290110766, −3.79069595488229119025126157581, −3.58939469321543617510404624497, −3.18925518687680797930743848234, −3.13982413445210366364727712954, −3.06145506609494746425749564675, −3.00305953409890745961770835555, −2.63566956795162863989981164754, −2.40170104893698764821493491996, −2.27307116491759756972028624756, −2.08434337102373577935838307825, −1.76750952452637660642121854156, −1.71764616959172982873608564254, −1.32003259419381214007087611847, −1.06868189256490683637321359464, −0.881377464071613570756135362286, −0.47052165959834304110203526145, −0.46809132052956985688790627430, 0.46809132052956985688790627430, 0.47052165959834304110203526145, 0.881377464071613570756135362286, 1.06868189256490683637321359464, 1.32003259419381214007087611847, 1.71764616959172982873608564254, 1.76750952452637660642121854156, 2.08434337102373577935838307825, 2.27307116491759756972028624756, 2.40170104893698764821493491996, 2.63566956795162863989981164754, 3.00305953409890745961770835555, 3.06145506609494746425749564675, 3.13982413445210366364727712954, 3.18925518687680797930743848234, 3.58939469321543617510404624497, 3.79069595488229119025126157581, 3.91104710240959013538290110766, 4.06672036594375677088553941964, 4.14194120348923618317526728211, 4.53619344600921363840534939668, 4.56517972860610334833553134219, 5.02979934964389408215927178843, 5.06075910948859889058815137531, 5.08832947017555354882269059197

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