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  + 2.45·2-s − 3-s + 4.02·4-s + 0.940·5-s − 2.45·6-s − 2.20·7-s + 4.97·8-s + 9-s + 2.30·10-s − 2.19·11-s − 4.02·12-s + 1.05·13-s − 5.41·14-s − 0.940·15-s + 4.16·16-s − 17-s + 2.45·18-s + 2.34·19-s + 3.78·20-s + 2.20·21-s − 5.38·22-s − 4.57·23-s − 4.97·24-s − 4.11·25-s + 2.59·26-s − 27-s − 8.87·28-s + ⋯
L(s)  = 1  + 1.73·2-s − 0.577·3-s + 2.01·4-s + 0.420·5-s − 1.00·6-s − 0.833·7-s + 1.76·8-s + 0.333·9-s + 0.730·10-s − 0.660·11-s − 1.16·12-s + 0.292·13-s − 1.44·14-s − 0.242·15-s + 1.04·16-s − 0.242·17-s + 0.578·18-s + 0.536·19-s + 0.846·20-s + 0.480·21-s − 1.14·22-s − 0.953·23-s − 1.01·24-s − 0.823·25-s + 0.508·26-s − 0.192·27-s − 1.67·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

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 \( 1 - 2.45T + 2T^{2} \)
5 \( 1 - 0.940T + 5T^{2} \)
7 \( 1 + 2.20T + 7T^{2} \)
11 \( 1 + 2.19T + 11T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
23 \( 1 + 4.57T + 23T^{2} \)
29 \( 1 + 0.413T + 29T^{2} \)
31 \( 1 + 5.18T + 31T^{2} \)
37 \( 1 + 4.26T + 37T^{2} \)
41 \( 1 + 1.99T + 41T^{2} \)
43 \( 1 - 5.39T + 43T^{2} \)
47 \( 1 - 0.440T + 47T^{2} \)
53 \( 1 - 2.47T + 53T^{2} \)
59 \( 1 + 6.59T + 59T^{2} \)
61 \( 1 + 7.41T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 4.61T + 71T^{2} \)
73 \( 1 + 16.5T + 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 - 5.79T + 83T^{2} \)
89 \( 1 + 0.0466T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−7.20055464654104630757759300878, −6.37327086389044918960067232225, −5.96729138475968620506380185304, −5.44564110930225799524168152785, −4.76590229949966091599022128240, −3.92332297033576902354624189403, −3.36041442513747194346174753637, −2.48793601725034665271303630822, −1.67482977704844313402489572025, 0, 1.67482977704844313402489572025, 2.48793601725034665271303630822, 3.36041442513747194346174753637, 3.92332297033576902354624189403, 4.76590229949966091599022128240, 5.44564110930225799524168152785, 5.96729138475968620506380185304, 6.37327086389044918960067232225, 7.20055464654104630757759300878

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