Properties

Degree 2
Conductor $ 2^{3} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.29·3-s − 1.23·5-s − 2.46·7-s − 1.31·9-s − 0.0568·11-s + 4.73·13-s + 1.60·15-s − 17-s − 0.278·19-s + 3.19·21-s + 1.06·23-s − 3.47·25-s + 5.60·27-s − 6.61·29-s + 4.85·31-s + 0.0737·33-s + 3.03·35-s + 0.645·37-s − 6.14·39-s + 3.28·41-s − 2.78·43-s + 1.62·45-s − 4.10·47-s − 0.941·49-s + 1.29·51-s + 9.36·53-s + 0.0702·55-s + ⋯
L(s)  = 1  − 0.748·3-s − 0.552·5-s − 0.930·7-s − 0.439·9-s − 0.0171·11-s + 1.31·13-s + 0.413·15-s − 0.242·17-s − 0.0639·19-s + 0.696·21-s + 0.222·23-s − 0.695·25-s + 1.07·27-s − 1.22·29-s + 0.872·31-s + 0.0128·33-s + 0.513·35-s + 0.106·37-s − 0.983·39-s + 0.513·41-s − 0.425·43-s + 0.242·45-s − 0.599·47-s − 0.134·49-s + 0.181·51-s + 1.28·53-s + 0.00946·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8024,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 + T \)
good3 \( 1 + 1.29T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 + 0.0568T + 11T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
19 \( 1 + 0.278T + 19T^{2} \)
23 \( 1 - 1.06T + 23T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 - 4.85T + 31T^{2} \)
37 \( 1 - 0.645T + 37T^{2} \)
41 \( 1 - 3.28T + 41T^{2} \)
43 \( 1 + 2.78T + 43T^{2} \)
47 \( 1 + 4.10T + 47T^{2} \)
53 \( 1 - 9.36T + 53T^{2} \)
61 \( 1 - 9.88T + 61T^{2} \)
67 \( 1 + 3.10T + 67T^{2} \)
71 \( 1 - 0.938T + 71T^{2} \)
73 \( 1 - 5.16T + 73T^{2} \)
79 \( 1 + 1.50T + 79T^{2} \)
83 \( 1 - 0.594T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + 3.06T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.40744046837946770241362338036, −6.60512751554959561994653973407, −6.12728643967567524629933308632, −5.58140095623579322057791639825, −4.69734386192750549097483440893, −3.77384114885183921750911428270, −3.33595532756009198216104872768, −2.26689342460616958314636435429, −0.947294511350439989500358879279, 0, 0.947294511350439989500358879279, 2.26689342460616958314636435429, 3.33595532756009198216104872768, 3.77384114885183921750911428270, 4.69734386192750549097483440893, 5.58140095623579322057791639825, 6.12728643967567524629933308632, 6.60512751554959561994653973407, 7.40744046837946770241362338036

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