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  + 1.70·2-s + 3-s + 0.896·4-s + 1.19·5-s + 1.70·6-s + 0.556·7-s − 1.87·8-s + 9-s + 2.04·10-s − 5.84·11-s + 0.896·12-s + 1.11·13-s + 0.947·14-s + 1.19·15-s − 4.98·16-s − 17-s + 1.70·18-s + 5.86·19-s + 1.07·20-s + 0.556·21-s − 9.94·22-s − 8.26·23-s − 1.87·24-s − 3.56·25-s + 1.90·26-s + 27-s + 0.499·28-s + ⋯
L(s)  = 1  + 1.20·2-s + 0.577·3-s + 0.448·4-s + 0.536·5-s + 0.694·6-s + 0.210·7-s − 0.663·8-s + 0.333·9-s + 0.645·10-s − 1.76·11-s + 0.258·12-s + 0.310·13-s + 0.253·14-s + 0.309·15-s − 1.24·16-s − 0.242·17-s + 0.401·18-s + 1.34·19-s + 0.240·20-s + 0.121·21-s − 2.12·22-s − 1.72·23-s − 0.383·24-s − 0.712·25-s + 0.373·26-s + 0.192·27-s + 0.0943·28-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 \( 1 - 1.70T + 2T^{2} \)
5 \( 1 - 1.19T + 5T^{2} \)
7 \( 1 - 0.556T + 7T^{2} \)
11 \( 1 + 5.84T + 11T^{2} \)
13 \( 1 - 1.11T + 13T^{2} \)
19 \( 1 - 5.86T + 19T^{2} \)
23 \( 1 + 8.26T + 23T^{2} \)
29 \( 1 + 0.910T + 29T^{2} \)
31 \( 1 - 8.49T + 31T^{2} \)
37 \( 1 + 6.72T + 37T^{2} \)
41 \( 1 + 0.589T + 41T^{2} \)
43 \( 1 + 2.41T + 43T^{2} \)
47 \( 1 + 9.83T + 47T^{2} \)
53 \( 1 + 2.20T + 53T^{2} \)
59 \( 1 - 3.63T + 59T^{2} \)
61 \( 1 + 5.37T + 61T^{2} \)
67 \( 1 + 4.71T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 5.95T + 73T^{2} \)
79 \( 1 - 9.93T + 79T^{2} \)
83 \( 1 - 4.17T + 83T^{2} \)
89 \( 1 - 9.98T + 89T^{2} \)
97 \( 1 - 4.02T + 97T^{2} \)
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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

−7.62208126374198313761501201799, −6.49834756784865157938460830518, −5.97264028302790457049556226053, −5.16948746700882164707584848973, −4.84779151265482076653003610289, −3.86024886680050282842660463421, −3.16749040202264973018365576979, −2.51704637052828820834291999184, −1.70717785744540917349280316656, 0, 1.70717785744540917349280316656, 2.51704637052828820834291999184, 3.16749040202264973018365576979, 3.86024886680050282842660463421, 4.84779151265482076653003610289, 5.16948746700882164707584848973, 5.97264028302790457049556226053, 6.49834756784865157938460830518, 7.62208126374198313761501201799

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