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  − 2.58·2-s + 3-s + 4.67·4-s + 2.37·5-s − 2.58·6-s − 1.51·7-s − 6.90·8-s + 9-s − 6.12·10-s + 0.0286·11-s + 4.67·12-s − 2.28·13-s + 3.90·14-s + 2.37·15-s + 8.47·16-s − 17-s − 2.58·18-s + 3.73·19-s + 11.0·20-s − 1.51·21-s − 0.0740·22-s + 0.0338·23-s − 6.90·24-s + 0.629·25-s + 5.89·26-s + 27-s − 7.06·28-s + ⋯
L(s)  = 1  − 1.82·2-s + 0.577·3-s + 2.33·4-s + 1.06·5-s − 1.05·6-s − 0.571·7-s − 2.43·8-s + 0.333·9-s − 1.93·10-s + 0.00864·11-s + 1.34·12-s − 0.632·13-s + 1.04·14-s + 0.612·15-s + 2.11·16-s − 0.242·17-s − 0.608·18-s + 0.857·19-s + 2.47·20-s − 0.330·21-s − 0.0157·22-s + 0.00705·23-s − 1.40·24-s + 0.125·25-s + 1.15·26-s + 0.192·27-s − 1.33·28-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 \( 1 + 2.58T + 2T^{2} \)
5 \( 1 - 2.37T + 5T^{2} \)
7 \( 1 + 1.51T + 7T^{2} \)
11 \( 1 - 0.0286T + 11T^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
19 \( 1 - 3.73T + 19T^{2} \)
23 \( 1 - 0.0338T + 23T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 - 5.40T + 31T^{2} \)
37 \( 1 + 4.32T + 37T^{2} \)
41 \( 1 - 9.63T + 41T^{2} \)
43 \( 1 + 6.66T + 43T^{2} \)
47 \( 1 + 7.80T + 47T^{2} \)
53 \( 1 + 4.66T + 53T^{2} \)
59 \( 1 + 6.95T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 - 6.50T + 73T^{2} \)
79 \( 1 - 7.88T + 79T^{2} \)
83 \( 1 + 6.05T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−7.67944404950283513131728065451, −7.02145015289742420863419001714, −6.40985286718794585102397585822, −5.77008192548688288504581293794, −4.77922210634458637516550149206, −3.41792616901272457118068825100, −2.70861467015625811457556328967, −2.00174487909884486182711250773, −1.27874478019351404392992755590, 0, 1.27874478019351404392992755590, 2.00174487909884486182711250773, 2.70861467015625811457556328967, 3.41792616901272457118068825100, 4.77922210634458637516550149206, 5.77008192548688288504581293794, 6.40985286718794585102397585822, 7.02145015289742420863419001714, 7.67944404950283513131728065451

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