Properties

Label 24-1216e12-1.1-c1e12-0-3
Degree $24$
Conductor $1.045\times 10^{37}$
Sign $1$
Analytic cond. $7.02308\times 10^{11}$
Root an. cond. $3.11605$
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  + 60·25-s − 132·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + 257-s + ⋯
L(s)  = 1  + 12·25-s − 12·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + 0.0623·257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(7.02308\times 10^{11}\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(29.24674795\)
\(L(\frac12)\) \(\approx\) \(29.24674795\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( ( 1 + p T^{2} )^{6} \)
good3 \( ( 1 + 22 T^{6} + p^{6} T^{12} )^{2} \)
5 \( ( 1 - p T^{2} )^{12} \)
7 \( ( 1 - 682 T^{6} + p^{6} T^{12} )^{2} \)
11 \( ( 1 + p T^{2} )^{12} \)
13 \( ( 1 + 4318 T^{6} + p^{6} T^{12} )^{2} \)
17 \( ( 1 - 138 T^{3} + p^{3} T^{6} )^{4} \)
23 \( ( 1 + 5942 T^{6} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 33986 T^{6} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + p T^{2} )^{12} \)
37 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{6} \)
41 \( ( 1 - p T^{2} )^{12} \)
43 \( ( 1 + p T^{2} )^{12} \)
47 \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{6} \)
53 \( ( 1 + 100078 T^{6} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 163834 T^{6} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - p T^{2} )^{12} \)
67 \( ( 1 + 268598 T^{6} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + p T^{2} )^{12} \)
73 \( ( 1 - 394 T^{3} + p^{3} T^{6} )^{4} \)
79 \( ( 1 + p T^{2} )^{12} \)
83 \( ( 1 + p T^{2} )^{12} \)
89 \( ( 1 - p T^{2} )^{12} \)
97 \( ( 1 - p T^{2} )^{12} \)
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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

−3.01055252533244251913754319744, −2.97389607941044434835533942242, −2.85352720144259483705445487312, −2.73827443079808372802454640087, −2.72759647693396293446416429936, −2.68627560934883496493254520034, −2.59854784271875324489770152520, −2.51054089492660453812627288728, −2.48814582241437693558831417997, −2.30676200482846665774037759052, −2.09017692081194110916467871165, −2.05308892410715951515805431451, −1.73112876918739201559841519292, −1.69041739654494811522173149503, −1.52248119877635404466962452961, −1.47906090322538787854137960859, −1.42381966829404287055727586117, −1.21310877272745023636475915826, −1.02530149836196833590649164211, −1.00442505296506674979221834573, −0.963458208018100116838881629643, −0.902363341602461692473437675316, −0.46324710390559103497528091274, −0.43552011424824182956852510360, −0.35649047348992046243818960061, 0.35649047348992046243818960061, 0.43552011424824182956852510360, 0.46324710390559103497528091274, 0.902363341602461692473437675316, 0.963458208018100116838881629643, 1.00442505296506674979221834573, 1.02530149836196833590649164211, 1.21310877272745023636475915826, 1.42381966829404287055727586117, 1.47906090322538787854137960859, 1.52248119877635404466962452961, 1.69041739654494811522173149503, 1.73112876918739201559841519292, 2.05308892410715951515805431451, 2.09017692081194110916467871165, 2.30676200482846665774037759052, 2.48814582241437693558831417997, 2.51054089492660453812627288728, 2.59854784271875324489770152520, 2.68627560934883496493254520034, 2.72759647693396293446416429936, 2.73827443079808372802454640087, 2.85352720144259483705445487312, 2.97389607941044434835533942242, 3.01055252533244251913754319744

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

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