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  − 0.943·2-s + 3-s − 1.10·4-s + 3.50·5-s − 0.943·6-s − 2.32·7-s + 2.93·8-s + 9-s − 3.31·10-s − 5.01·11-s − 1.10·12-s + 1.00·13-s + 2.19·14-s + 3.50·15-s − 0.551·16-s − 17-s − 0.943·18-s + 3.26·19-s − 3.89·20-s − 2.32·21-s + 4.73·22-s − 3.55·23-s + 2.93·24-s + 7.31·25-s − 0.948·26-s + 27-s + 2.58·28-s + ⋯
L(s)  = 1  − 0.667·2-s + 0.577·3-s − 0.554·4-s + 1.56·5-s − 0.385·6-s − 0.879·7-s + 1.03·8-s + 0.333·9-s − 1.04·10-s − 1.51·11-s − 0.320·12-s + 0.278·13-s + 0.587·14-s + 0.905·15-s − 0.137·16-s − 0.242·17-s − 0.222·18-s + 0.749·19-s − 0.870·20-s − 0.508·21-s + 1.00·22-s − 0.740·23-s + 0.599·24-s + 1.46·25-s − 0.186·26-s + 0.192·27-s + 0.487·28-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 \( 1 + 0.943T + 2T^{2} \)
5 \( 1 - 3.50T + 5T^{2} \)
7 \( 1 + 2.32T + 7T^{2} \)
11 \( 1 + 5.01T + 11T^{2} \)
13 \( 1 - 1.00T + 13T^{2} \)
19 \( 1 - 3.26T + 19T^{2} \)
23 \( 1 + 3.55T + 23T^{2} \)
29 \( 1 + 4.86T + 29T^{2} \)
31 \( 1 - 2.30T + 31T^{2} \)
37 \( 1 + 1.16T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 - 9.68T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 5.16T + 59T^{2} \)
61 \( 1 + 1.49T + 61T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 + 6.84T + 71T^{2} \)
73 \( 1 + 1.60T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 8.66T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 6.59T + 97T^{2} \)
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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

−7.56071447367216260853250845756, −7.02096857948676843956782227191, −5.92976350236946294056795027935, −5.58021388525945989020539079224, −4.76646398781632102957670603375, −3.79478947326828372248557857544, −2.85259075618264400506224882854, −2.21192542393273280662814814892, −1.28527487009159258968087057841, 0, 1.28527487009159258968087057841, 2.21192542393273280662814814892, 2.85259075618264400506224882854, 3.79478947326828372248557857544, 4.76646398781632102957670603375, 5.58021388525945989020539079224, 5.92976350236946294056795027935, 7.02096857948676843956782227191, 7.56071447367216260853250845756

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