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  − 2.60·2-s + 3-s + 4.81·4-s − 2.83·5-s − 2.60·6-s + 1.53·7-s − 7.33·8-s + 9-s + 7.40·10-s − 5.14·11-s + 4.81·12-s − 5.76·13-s − 4.01·14-s − 2.83·15-s + 9.53·16-s − 17-s − 2.60·18-s + 7.46·19-s − 13.6·20-s + 1.53·21-s + 13.4·22-s + 3.10·23-s − 7.33·24-s + 3.04·25-s + 15.0·26-s + 27-s + 7.40·28-s + ⋯
L(s)  = 1  − 1.84·2-s + 0.577·3-s + 2.40·4-s − 1.26·5-s − 1.06·6-s + 0.581·7-s − 2.59·8-s + 0.333·9-s + 2.34·10-s − 1.55·11-s + 1.38·12-s − 1.59·13-s − 1.07·14-s − 0.732·15-s + 2.38·16-s − 0.242·17-s − 0.615·18-s + 1.71·19-s − 3.05·20-s + 0.335·21-s + 2.86·22-s + 0.647·23-s − 1.49·24-s + 0.608·25-s + 2.94·26-s + 0.192·27-s + 1.39·28-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 \( 1 + 2.60T + 2T^{2} \)
5 \( 1 + 2.83T + 5T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 + 5.14T + 11T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
19 \( 1 - 7.46T + 19T^{2} \)
23 \( 1 - 3.10T + 23T^{2} \)
29 \( 1 - 3.18T + 29T^{2} \)
31 \( 1 - 2.43T + 31T^{2} \)
37 \( 1 + 8.32T + 37T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 - 3.70T + 43T^{2} \)
47 \( 1 + 1.15T + 47T^{2} \)
53 \( 1 + 3.51T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 0.108T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 - 0.363T + 71T^{2} \)
73 \( 1 - 8.47T + 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 + 2.62T + 83T^{2} \)
89 \( 1 - 3.19T + 89T^{2} \)
97 \( 1 - 9.70T + 97T^{2} \)
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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

−7.76161027155864172698439423785, −7.29776899168728977696559772059, −6.74337481281536288721371655873, −5.25341628505207489030836253502, −4.85315125256329011911861168996, −3.50133746707785400695570202891, −2.76209049143296870412405249998, −2.16632255921102830888699893840, −0.932247115006207192406969078461, 0, 0.932247115006207192406969078461, 2.16632255921102830888699893840, 2.76209049143296870412405249998, 3.50133746707785400695570202891, 4.85315125256329011911861168996, 5.25341628505207489030836253502, 6.74337481281536288721371655873, 7.29776899168728977696559772059, 7.76161027155864172698439423785

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