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  − 0.0333·2-s + 3-s − 1.99·4-s − 1.35·5-s − 0.0333·6-s − 2.46·7-s + 0.133·8-s + 9-s + 0.0452·10-s + 0.516·11-s − 1.99·12-s + 4.34·13-s + 0.0820·14-s − 1.35·15-s + 3.99·16-s − 17-s − 0.0333·18-s − 3.60·19-s + 2.71·20-s − 2.46·21-s − 0.0172·22-s − 3.19·23-s + 0.133·24-s − 3.16·25-s − 0.144·26-s + 27-s + 4.92·28-s + ⋯
L(s)  = 1  − 0.0235·2-s + 0.577·3-s − 0.999·4-s − 0.606·5-s − 0.0136·6-s − 0.930·7-s + 0.0471·8-s + 0.333·9-s + 0.0142·10-s + 0.155·11-s − 0.577·12-s + 1.20·13-s + 0.0219·14-s − 0.350·15-s + 0.998·16-s − 0.242·17-s − 0.00785·18-s − 0.826·19-s + 0.605·20-s − 0.537·21-s − 0.00367·22-s − 0.665·23-s + 0.0272·24-s − 0.632·25-s − 0.0284·26-s + 0.192·27-s + 0.930·28-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 \( 1 + 0.0333T + 2T^{2} \)
5 \( 1 + 1.35T + 5T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 - 0.516T + 11T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 + 3.19T + 23T^{2} \)
29 \( 1 - 6.91T + 29T^{2} \)
31 \( 1 + 0.288T + 31T^{2} \)
37 \( 1 + 4.39T + 37T^{2} \)
41 \( 1 + 3.00T + 41T^{2} \)
43 \( 1 - 7.40T + 43T^{2} \)
47 \( 1 - 2.99T + 47T^{2} \)
53 \( 1 - 3.25T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 8.97T + 61T^{2} \)
67 \( 1 + 0.00314T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 + 5.21T + 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
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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

−7.67567796151084467084482549724, −6.76855979907837258095218581004, −6.18735934567390930811002664614, −5.38075456930231098121280729280, −4.32379299869641457581444919155, −3.89650450642840456888687588899, −3.38179577374677303363015698948, −2.35052128638404815352684958015, −1.08128136158049455557082829874, 0, 1.08128136158049455557082829874, 2.35052128638404815352684958015, 3.38179577374677303363015698948, 3.89650450642840456888687588899, 4.32379299869641457581444919155, 5.38075456930231098121280729280, 6.18735934567390930811002664614, 6.76855979907837258095218581004, 7.67567796151084467084482549724

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