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  − 1.50·2-s + 3-s + 0.255·4-s − 0.915·5-s − 1.50·6-s − 3.11·7-s + 2.61·8-s + 9-s + 1.37·10-s + 1.83·11-s + 0.255·12-s − 4.69·13-s + 4.67·14-s − 0.915·15-s − 4.44·16-s − 17-s − 1.50·18-s + 4.81·19-s − 0.233·20-s − 3.11·21-s − 2.75·22-s − 2.49·23-s + 2.61·24-s − 4.16·25-s + 7.05·26-s + 27-s − 0.795·28-s + ⋯
L(s)  = 1  − 1.06·2-s + 0.577·3-s + 0.127·4-s − 0.409·5-s − 0.613·6-s − 1.17·7-s + 0.926·8-s + 0.333·9-s + 0.434·10-s + 0.552·11-s + 0.0737·12-s − 1.30·13-s + 1.24·14-s − 0.236·15-s − 1.11·16-s − 0.242·17-s − 0.353·18-s + 1.10·19-s − 0.0523·20-s − 0.679·21-s − 0.586·22-s − 0.520·23-s + 0.534·24-s − 0.832·25-s + 1.38·26-s + 0.192·27-s − 0.150·28-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 \( 1 + 1.50T + 2T^{2} \)
5 \( 1 + 0.915T + 5T^{2} \)
7 \( 1 + 3.11T + 7T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 + 4.69T + 13T^{2} \)
19 \( 1 - 4.81T + 19T^{2} \)
23 \( 1 + 2.49T + 23T^{2} \)
29 \( 1 - 4.17T + 29T^{2} \)
31 \( 1 - 7.38T + 31T^{2} \)
37 \( 1 + 5.82T + 37T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 + 3.88T + 43T^{2} \)
47 \( 1 - 3.13T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 3.86T + 59T^{2} \)
61 \( 1 - 4.04T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 4.17T + 71T^{2} \)
73 \( 1 - 3.90T + 73T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + 2.87T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 5.66T + 97T^{2} \)
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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

−7.52023654785945207490989935430, −7.12283655989925926070419749981, −6.44591769677488425514028649133, −5.39415609935887084279679161754, −4.48248909477998272235684615051, −3.85080990213313800637943884830, −2.97701417156614626286596371914, −2.18103360127943163720284141678, −0.993350212763072398355099512923, 0, 0.993350212763072398355099512923, 2.18103360127943163720284141678, 2.97701417156614626286596371914, 3.85080990213313800637943884830, 4.48248909477998272235684615051, 5.39415609935887084279679161754, 6.44591769677488425514028649133, 7.12283655989925926070419749981, 7.52023654785945207490989935430

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