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  + 0.561·3-s − 2.34·5-s − 3.24·7-s − 2.68·9-s + 0.117·11-s + 0.0710·13-s − 1.31·15-s − 17-s + 7.74·19-s − 1.82·21-s + 3.00·23-s + 0.477·25-s − 3.19·27-s + 7.74·29-s + 0.790·31-s + 0.0658·33-s + 7.59·35-s + 5.26·37-s + 0.0399·39-s − 4.64·41-s + 5.78·43-s + 6.28·45-s − 2.92·47-s + 3.53·49-s − 0.561·51-s − 4.98·53-s − 0.274·55-s + ⋯
L(s)  = 1  + 0.324·3-s − 1.04·5-s − 1.22·7-s − 0.894·9-s + 0.0353·11-s + 0.0196·13-s − 0.339·15-s − 0.242·17-s + 1.77·19-s − 0.398·21-s + 0.626·23-s + 0.0954·25-s − 0.614·27-s + 1.43·29-s + 0.141·31-s + 0.0114·33-s + 1.28·35-s + 0.864·37-s + 0.00639·39-s − 0.724·41-s + 0.881·43-s + 0.936·45-s − 0.426·47-s + 0.505·49-s − 0.0786·51-s − 0.684·53-s − 0.0369·55-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 \( 1 - 0.561T + 3T^{2} \)
5 \( 1 + 2.34T + 5T^{2} \)
7 \( 1 + 3.24T + 7T^{2} \)
11 \( 1 - 0.117T + 11T^{2} \)
13 \( 1 - 0.0710T + 13T^{2} \)
19 \( 1 - 7.74T + 19T^{2} \)
23 \( 1 - 3.00T + 23T^{2} \)
29 \( 1 - 7.74T + 29T^{2} \)
31 \( 1 - 0.790T + 31T^{2} \)
37 \( 1 - 5.26T + 37T^{2} \)
41 \( 1 + 4.64T + 41T^{2} \)
43 \( 1 - 5.78T + 43T^{2} \)
47 \( 1 + 2.92T + 47T^{2} \)
53 \( 1 + 4.98T + 53T^{2} \)
61 \( 1 - 3.39T + 61T^{2} \)
67 \( 1 - 4.95T + 67T^{2} \)
71 \( 1 - 4.72T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 6.93T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 - 1.43T + 97T^{2} \)
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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

−7.53597208406161658068459452544, −6.86656128193105207047203776882, −6.18606537638624216108709282876, −5.41940100775627991612250406516, −4.58304662855159421607828166236, −3.67096137203390921089473495887, −3.13159216346556993608769824675, −2.62579450909403994893928040651, −1.02263307443737199392330282849, 0, 1.02263307443737199392330282849, 2.62579450909403994893928040651, 3.13159216346556993608769824675, 3.67096137203390921089473495887, 4.58304662855159421607828166236, 5.41940100775627991612250406516, 6.18606537638624216108709282876, 6.86656128193105207047203776882, 7.53597208406161658068459452544

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