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Properties

Label 2-336e2-1.1-c1-0-59
Degree $2$
Conductor $112896$
Sign $-1$
Analytic cond. $901.479$
Root an. cond. $30.0246$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 4·13-s − 2·17-s + 11·25-s + 4·29-s + 12·37-s − 10·41-s + 4·53-s − 12·61-s − 16·65-s + 6·73-s + 8·85-s + 10·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.10·13-s − 0.485·17-s + 11/5·25-s + 0.742·29-s + 1.97·37-s − 1.56·41-s + 0.549·53-s − 1.53·61-s − 1.98·65-s + 0.702·73-s + 0.867·85-s + 1.05·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(901.479\)
Root analytic conductor: \(30.0246\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 112896,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.76206934043810, −13.38857005768349, −12.86307025266194, −12.32469286075744, −11.82296664623991, −11.52546307265640, −11.02466042502335, −10.63066413629750, −10.10604120115103, −9.247625409888427, −8.911929482259566, −8.259247086880951, −8.030265938733482, −7.534309473673754, −6.856597054753391, −6.476226772137012, −5.896724288871156, −5.055409874450600, −4.566752594435659, −4.080911075254383, −3.591688815960497, −3.079195687458156, −2.414928500841280, −1.395278222161286, −0.7780548646881631, 0, 0.7780548646881631, 1.395278222161286, 2.414928500841280, 3.079195687458156, 3.591688815960497, 4.080911075254383, 4.566752594435659, 5.055409874450600, 5.896724288871156, 6.476226772137012, 6.856597054753391, 7.534309473673754, 8.030265938733482, 8.259247086880951, 8.911929482259566, 9.247625409888427, 10.10604120115103, 10.63066413629750, 11.02466042502335, 11.52546307265640, 11.82296664623991, 12.32469286075744, 12.86307025266194, 13.38857005768349, 13.76206934043810

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