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  − 2.80·2-s + 3-s + 5.88·4-s − 2.90·5-s − 2.80·6-s − 1.28·7-s − 10.9·8-s + 9-s + 8.16·10-s + 2.87·11-s + 5.88·12-s + 0.133·13-s + 3.61·14-s − 2.90·15-s + 18.8·16-s − 17-s − 2.80·18-s − 3.66·19-s − 17.1·20-s − 1.28·21-s − 8.06·22-s − 6.74·23-s − 10.9·24-s + 3.45·25-s − 0.374·26-s + 27-s − 7.58·28-s + ⋯
L(s)  = 1  − 1.98·2-s + 0.577·3-s + 2.94·4-s − 1.30·5-s − 1.14·6-s − 0.486·7-s − 3.85·8-s + 0.333·9-s + 2.58·10-s + 0.865·11-s + 1.69·12-s + 0.0369·13-s + 0.966·14-s − 0.750·15-s + 4.71·16-s − 0.242·17-s − 0.661·18-s − 0.839·19-s − 3.82·20-s − 0.281·21-s − 1.71·22-s − 1.40·23-s − 2.22·24-s + 0.691·25-s − 0.0733·26-s + 0.192·27-s − 1.43·28-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 \( 1 + 2.80T + 2T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 - 2.87T + 11T^{2} \)
13 \( 1 - 0.133T + 13T^{2} \)
19 \( 1 + 3.66T + 19T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + 3.28T + 29T^{2} \)
31 \( 1 - 8.99T + 31T^{2} \)
37 \( 1 - 2.72T + 37T^{2} \)
41 \( 1 - 6.71T + 41T^{2} \)
43 \( 1 - 5.12T + 43T^{2} \)
47 \( 1 - 3.31T + 47T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 - 1.13T + 61T^{2} \)
67 \( 1 + 5.26T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 5.39T + 79T^{2} \)
83 \( 1 + 9.57T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + 8.78T + 97T^{2} \)
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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

−7.79990645351245544122566793245, −7.12410228790419701343655695605, −6.47656329963373076970938067350, −5.96124928636607966204755751024, −4.27515999038111862897060547484, −3.69256463691033108603300688063, −2.80278770230908203118817917176, −2.04249970815033471440505177340, −0.959414170124880857611425430628, 0, 0.959414170124880857611425430628, 2.04249970815033471440505177340, 2.80278770230908203118817917176, 3.69256463691033108603300688063, 4.27515999038111862897060547484, 5.96124928636607966204755751024, 6.47656329963373076970938067350, 7.12410228790419701343655695605, 7.79990645351245544122566793245

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