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Properties

Label 2-304e2-1.1-c1-0-28
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  − 3-s − 4·5-s + 4·7-s − 2·9-s + 5·11-s + 4·13-s + 4·15-s − 2·17-s − 4·21-s + 11·25-s + 5·27-s − 4·29-s − 8·31-s − 5·33-s − 16·35-s − 8·37-s − 4·39-s − 3·41-s + 8·43-s + 8·45-s + 12·47-s + 9·49-s + 2·51-s − 12·53-s − 20·55-s − 3·59-s + 4·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1.51·7-s − 2/3·9-s + 1.50·11-s + 1.10·13-s + 1.03·15-s − 0.485·17-s − 0.872·21-s + 11/5·25-s + 0.962·27-s − 0.742·29-s − 1.43·31-s − 0.870·33-s − 2.70·35-s − 1.31·37-s − 0.640·39-s − 0.468·41-s + 1.21·43-s + 1.19·45-s + 1.75·47-s + 9/7·49-s + 0.280·51-s − 1.64·53-s − 2.69·55-s − 0.390·59-s + 0.512·61-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 \( 1 + T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−14.20652531788603, −13.80448926918585, −12.96151771251783, −12.17902863847094, −12.11017063034179, −11.58479130694855, −11.10374121937438, −10.88852548288310, −10.66929526193873, −9.274906866314656, −8.967700996928894, −8.613375864485324, −7.969519971140669, −7.686131975812548, −6.982156450748576, −6.558128699091287, −5.853751682383429, −5.249147031708537, −4.749760138390817, −4.050172353183900, −3.814332061867748, −3.270575466971468, −2.180270921929089, −1.432216590774753, −0.8499873396253441, 0, 0.8499873396253441, 1.432216590774753, 2.180270921929089, 3.270575466971468, 3.814332061867748, 4.050172353183900, 4.749760138390817, 5.249147031708537, 5.853751682383429, 6.558128699091287, 6.982156450748576, 7.686131975812548, 7.969519971140669, 8.613375864485324, 8.967700996928894, 9.274906866314656, 10.66929526193873, 10.88852548288310, 11.10374121937438, 11.58479130694855, 12.11017063034179, 12.17902863847094, 12.96151771251783, 13.80448926918585, 14.20652531788603

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