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  + 1.57·2-s + 3-s + 0.474·4-s + 0.371·5-s + 1.57·6-s − 2.37·7-s − 2.40·8-s + 9-s + 0.585·10-s + 3.67·11-s + 0.474·12-s + 4.51·13-s − 3.74·14-s + 0.371·15-s − 4.72·16-s − 17-s + 1.57·18-s − 3.53·19-s + 0.176·20-s − 2.37·21-s + 5.78·22-s − 4.93·23-s − 2.40·24-s − 4.86·25-s + 7.10·26-s + 27-s − 1.12·28-s + ⋯
L(s)  = 1  + 1.11·2-s + 0.577·3-s + 0.237·4-s + 0.166·5-s + 0.642·6-s − 0.899·7-s − 0.848·8-s + 0.333·9-s + 0.184·10-s + 1.10·11-s + 0.136·12-s + 1.25·13-s − 1.00·14-s + 0.0960·15-s − 1.18·16-s − 0.242·17-s + 0.370·18-s − 0.810·19-s + 0.0394·20-s − 0.519·21-s + 1.23·22-s − 1.02·23-s − 0.489·24-s − 0.972·25-s + 1.39·26-s + 0.192·27-s − 0.213·28-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 \( 1 - 1.57T + 2T^{2} \)
5 \( 1 - 0.371T + 5T^{2} \)
7 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
13 \( 1 - 4.51T + 13T^{2} \)
19 \( 1 + 3.53T + 19T^{2} \)
23 \( 1 + 4.93T + 23T^{2} \)
29 \( 1 + 8.06T + 29T^{2} \)
31 \( 1 + 3.71T + 31T^{2} \)
37 \( 1 + 4.99T + 37T^{2} \)
41 \( 1 - 6.37T + 41T^{2} \)
43 \( 1 + 5.79T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 4.55T + 59T^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 - 0.669T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 - 6.56T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−7.23192060313550318470005871230, −6.62157434438937341791327590862, −5.87977278188274883747058170521, −5.65746113030484400176456511285, −4.21443602426379665442753328802, −3.95105421373106983744636014470, −3.46810973697428754859437984329, −2.46166528138872083544301335993, −1.58145453748822349645364155693, 0, 1.58145453748822349645364155693, 2.46166528138872083544301335993, 3.46810973697428754859437984329, 3.95105421373106983744636014470, 4.21443602426379665442753328802, 5.65746113030484400176456511285, 5.87977278188274883747058170521, 6.62157434438937341791327590862, 7.23192060313550318470005871230

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