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Properties

Label 2-304e2-1.1-c1-0-34
Degree $2$
Conductor $92416$
Sign $-1$
Analytic cond. $737.945$
Root an. cond. $27.1651$
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 − 3·9-s − 4·13-s − 2·17-s + 11·25-s − 4·29-s + 12·37-s + 10·41-s − 12·45-s − 7·49-s − 4·53-s − 12·61-s − 16·65-s − 6·73-s + 9·81-s − 8·85-s − 10·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 12·117-s + ⋯
L(s)  = 1  + 1.78·5-s − 9-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 − 1.78·45-s − 49-s − 0.549·53-s − 1.53·61-s − 1.98·65-s − 0.702·73-s + 81-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 + 1.10·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(737.945\)
Root analytic conductor: \(27.1651\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92416,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
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

−14.24693847314962, −13.50185575468499, −13.18379958372631, −12.76844640105760, −12.20673778408164, −11.55594251319234, −11.02663352730112, −10.68719451057981, −9.952173775387440, −9.587385197840610, −9.222486727680639, −8.815915020056202, −8.023685759289912, −7.555153216388725, −6.909629496814858, −6.135680286791673, −6.032219963786888, −5.489314232232484, −4.772104958782258, −4.483699354018548, −3.376524697438253, −2.724173740421614, −2.381126452102599, −1.806743933214244, −0.9639465490435756, 0, 0.9639465490435756, 1.806743933214244, 2.381126452102599, 2.724173740421614, 3.376524697438253, 4.483699354018548, 4.772104958782258, 5.489314232232484, 6.032219963786888, 6.135680286791673, 6.909629496814858, 7.555153216388725, 8.023685759289912, 8.815915020056202, 9.222486727680639, 9.587385197840610, 9.952173775387440, 10.68719451057981, 11.02663352730112, 11.55594251319234, 12.20673778408164, 12.76844640105760, 13.18379958372631, 13.50185575468499, 14.24693847314962

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