Properties

Label 2-156e2-1.1-c1-0-58
Degree $2$
Conductor $24336$
Sign $1$
Analytic cond. $194.323$
Root an. cond. $13.9400$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·11-s − 2·17-s − 4·19-s − 8·23-s − 25-s − 6·29-s + 8·31-s − 6·37-s − 6·41-s − 4·43-s − 7·49-s + 2·53-s + 8·55-s − 4·59-s − 2·61-s − 4·67-s − 8·71-s − 10·73-s + 8·79-s + 4·83-s + 4·85-s − 6·89-s + 8·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.20·11-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 49-s + 0.274·53-s + 1.07·55-s − 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.433·85-s − 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24336\)    =    \(2^{4} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(194.323\)
Root analytic conductor: \(13.9400\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{24336} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 24336,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.99757749866486, −15.32964808150714, −15.11246064320381, −14.40239851236580, −13.57334543565085, −13.39922089894708, −12.68868139298271, −12.00434434170607, −11.78464633445677, −11.03514914395727, −10.46360805572590, −10.10096349784598, −9.369781462314316, −8.525525288099009, −8.164389703758823, −7.745653438040241, −7.060134581080893, −6.369579148952002, −5.780054615909326, −5.006441000350352, −4.410848943757797, −3.832661908305993, −3.118838753052904, −2.307109076635718, −1.622834083754368, 0, 0, 1.622834083754368, 2.307109076635718, 3.118838753052904, 3.832661908305993, 4.410848943757797, 5.006441000350352, 5.780054615909326, 6.369579148952002, 7.060134581080893, 7.745653438040241, 8.164389703758823, 8.525525288099009, 9.369781462314316, 10.10096349784598, 10.46360805572590, 11.03514914395727, 11.78464633445677, 12.00434434170607, 12.68868139298271, 13.39922089894708, 13.57334543565085, 14.40239851236580, 15.11246064320381, 15.32964808150714, 15.99757749866486

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