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  − 0.901·2-s − 3-s − 1.18·4-s + 1.97·5-s + 0.901·6-s + 4.00·7-s + 2.87·8-s + 9-s − 1.78·10-s − 4.85·11-s + 1.18·12-s − 5.58·13-s − 3.60·14-s − 1.97·15-s − 0.214·16-s − 17-s − 0.901·18-s + 4.03·19-s − 2.34·20-s − 4.00·21-s + 4.37·22-s − 0.727·23-s − 2.87·24-s − 1.09·25-s + 5.03·26-s − 27-s − 4.75·28-s + ⋯
L(s)  = 1  − 0.637·2-s − 0.577·3-s − 0.593·4-s + 0.883·5-s + 0.367·6-s + 1.51·7-s + 1.01·8-s + 0.333·9-s − 0.563·10-s − 1.46·11-s + 0.342·12-s − 1.54·13-s − 0.964·14-s − 0.510·15-s − 0.0536·16-s − 0.242·17-s − 0.212·18-s + 0.926·19-s − 0.524·20-s − 0.873·21-s + 0.933·22-s − 0.151·23-s − 0.586·24-s − 0.218·25-s + 0.987·26-s − 0.192·27-s − 0.898·28-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 \( 1 + 0.901T + 2T^{2} \)
5 \( 1 - 1.97T + 5T^{2} \)
7 \( 1 - 4.00T + 7T^{2} \)
11 \( 1 + 4.85T + 11T^{2} \)
13 \( 1 + 5.58T + 13T^{2} \)
19 \( 1 - 4.03T + 19T^{2} \)
23 \( 1 + 0.727T + 23T^{2} \)
29 \( 1 - 6.57T + 29T^{2} \)
31 \( 1 + 2.80T + 31T^{2} \)
37 \( 1 - 1.01T + 37T^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 - 2.80T + 43T^{2} \)
47 \( 1 - 9.68T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 4.64T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 7.35T + 73T^{2} \)
79 \( 1 + 7.48T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 - 0.976T + 89T^{2} \)
97 \( 1 + 4.48T + 97T^{2} \)
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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

−7.63080099010939791877617665736, −7.12206541253840843941010192199, −5.79872256635239740302149197403, −5.35222822147221772737155523194, −4.79661633623638347951857319533, −4.35260731079776611028342861223, −2.76114020577218234610122466276, −2.02603884233775026881597331828, −1.15352899106214905267282555763, 0, 1.15352899106214905267282555763, 2.02603884233775026881597331828, 2.76114020577218234610122466276, 4.35260731079776611028342861223, 4.79661633623638347951857319533, 5.35222822147221772737155523194, 5.79872256635239740302149197403, 7.12206541253840843941010192199, 7.63080099010939791877617665736

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