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  + 2.78·2-s − 3-s + 5.77·4-s − 2.00·5-s − 2.78·6-s + 0.938·7-s + 10.5·8-s + 9-s − 5.59·10-s − 5.68·11-s − 5.77·12-s + 0.884·13-s + 2.61·14-s + 2.00·15-s + 17.8·16-s − 17-s + 2.78·18-s − 2.71·19-s − 11.5·20-s − 0.938·21-s − 15.8·22-s − 5.54·23-s − 10.5·24-s − 0.977·25-s + 2.46·26-s − 27-s + 5.42·28-s + ⋯
L(s)  = 1  + 1.97·2-s − 0.577·3-s + 2.88·4-s − 0.896·5-s − 1.13·6-s + 0.354·7-s + 3.72·8-s + 0.333·9-s − 1.76·10-s − 1.71·11-s − 1.66·12-s + 0.245·13-s + 0.699·14-s + 0.517·15-s + 4.45·16-s − 0.242·17-s + 0.657·18-s − 0.622·19-s − 2.59·20-s − 0.204·21-s − 3.38·22-s − 1.15·23-s − 2.15·24-s − 0.195·25-s + 0.483·26-s − 0.192·27-s + 1.02·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 - 2.78T + 2T^{2} \)
5 \( 1 + 2.00T + 5T^{2} \)
7 \( 1 - 0.938T + 7T^{2} \)
11 \( 1 + 5.68T + 11T^{2} \)
13 \( 1 - 0.884T + 13T^{2} \)
19 \( 1 + 2.71T + 19T^{2} \)
23 \( 1 + 5.54T + 23T^{2} \)
29 \( 1 + 2.01T + 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + 2.54T + 37T^{2} \)
41 \( 1 + 0.0735T + 41T^{2} \)
43 \( 1 + 9.11T + 43T^{2} \)
47 \( 1 - 6.75T + 47T^{2} \)
53 \( 1 + 6.52T + 53T^{2} \)
59 \( 1 - 4.86T + 59T^{2} \)
61 \( 1 - 5.04T + 61T^{2} \)
67 \( 1 + 4.00T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 8.53T + 79T^{2} \)
83 \( 1 - 3.13T + 83T^{2} \)
89 \( 1 + 9.82T + 89T^{2} \)
97 \( 1 + 8.29T + 97T^{2} \)
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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

−7.29512636391638086639523611143, −6.58886703995400330403380229514, −5.80864178529629829712180757618, −5.27348030758025954055858446296, −4.75555273870391425163363613409, −3.94681017900571089554125258238, −3.50596238869972607404731800989, −2.43269046501772608369785954473, −1.77146994999901311887202721471, 0, 1.77146994999901311887202721471, 2.43269046501772608369785954473, 3.50596238869972607404731800989, 3.94681017900571089554125258238, 4.75555273870391425163363613409, 5.27348030758025954055858446296, 5.80864178529629829712180757618, 6.58886703995400330403380229514, 7.29512636391638086639523611143

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