Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.375·2-s + 3-s − 1.85·4-s + 1.27·5-s − 0.375·6-s + 2.77·7-s + 1.44·8-s + 9-s − 0.479·10-s + 0.396·11-s − 1.85·12-s − 5.77·13-s − 1.04·14-s + 1.27·15-s + 3.17·16-s − 17-s − 0.375·18-s − 4.30·19-s − 2.37·20-s + 2.77·21-s − 0.149·22-s + 4.90·23-s + 1.44·24-s − 3.36·25-s + 2.16·26-s + 27-s − 5.16·28-s + ⋯
L(s)  = 1  − 0.265·2-s + 0.577·3-s − 0.929·4-s + 0.571·5-s − 0.153·6-s + 1.04·7-s + 0.512·8-s + 0.333·9-s − 0.151·10-s + 0.119·11-s − 0.536·12-s − 1.60·13-s − 0.278·14-s + 0.329·15-s + 0.793·16-s − 0.242·17-s − 0.0884·18-s − 0.988·19-s − 0.531·20-s + 0.605·21-s − 0.0317·22-s + 1.02·23-s + 0.295·24-s − 0.673·25-s + 0.424·26-s + 0.192·27-s − 0.975·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 0.375T + 2T^{2} \)
5 \( 1 - 1.27T + 5T^{2} \)
7 \( 1 - 2.77T + 7T^{2} \)
11 \( 1 - 0.396T + 11T^{2} \)
13 \( 1 + 5.77T + 13T^{2} \)
19 \( 1 + 4.30T + 19T^{2} \)
23 \( 1 - 4.90T + 23T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 + 4.07T + 37T^{2} \)
41 \( 1 - 6.26T + 41T^{2} \)
43 \( 1 + 5.27T + 43T^{2} \)
47 \( 1 - 8.22T + 47T^{2} \)
53 \( 1 + 6.53T + 53T^{2} \)
59 \( 1 + 1.83T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 2.21T + 83T^{2} \)
89 \( 1 + 9.04T + 89T^{2} \)
97 \( 1 - 19.1T + 97T^{2} \)
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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

−7.57524543635584836706797535716, −7.10715391114327012583016796081, −6.02202165468385284991625951177, −5.15285437792640864149683926451, −4.69403531308547613702529553124, −4.10675480739695827345550774582, −2.98255498654455115553163813772, −2.10953514221798185759483537616, −1.39264554737674375749785648090, 0, 1.39264554737674375749785648090, 2.10953514221798185759483537616, 2.98255498654455115553163813772, 4.10675480739695827345550774582, 4.69403531308547613702529553124, 5.15285437792640864149683926451, 6.02202165468385284991625951177, 7.10715391114327012583016796081, 7.57524543635584836706797535716

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