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  + 2.01·2-s + 3-s + 2.07·4-s − 1.34·5-s + 2.01·6-s − 0.512·7-s + 0.147·8-s + 9-s − 2.70·10-s + 0.645·11-s + 2.07·12-s − 0.0798·13-s − 1.03·14-s − 1.34·15-s − 3.84·16-s − 17-s + 2.01·18-s − 6.43·19-s − 2.77·20-s − 0.512·21-s + 1.30·22-s + 7.42·23-s + 0.147·24-s − 3.20·25-s − 0.161·26-s + 27-s − 1.06·28-s + ⋯
L(s)  = 1  + 1.42·2-s + 0.577·3-s + 1.03·4-s − 0.599·5-s + 0.823·6-s − 0.193·7-s + 0.0521·8-s + 0.333·9-s − 0.855·10-s + 0.194·11-s + 0.598·12-s − 0.0221·13-s − 0.276·14-s − 0.345·15-s − 0.962·16-s − 0.242·17-s + 0.475·18-s − 1.47·19-s − 0.621·20-s − 0.111·21-s + 0.277·22-s + 1.54·23-s + 0.0301·24-s − 0.640·25-s − 0.0315·26-s + 0.192·27-s − 0.200·28-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 \( 1 - 2.01T + 2T^{2} \)
5 \( 1 + 1.34T + 5T^{2} \)
7 \( 1 + 0.512T + 7T^{2} \)
11 \( 1 - 0.645T + 11T^{2} \)
13 \( 1 + 0.0798T + 13T^{2} \)
19 \( 1 + 6.43T + 19T^{2} \)
23 \( 1 - 7.42T + 23T^{2} \)
29 \( 1 - 4.78T + 29T^{2} \)
31 \( 1 - 3.55T + 31T^{2} \)
37 \( 1 + 6.33T + 37T^{2} \)
41 \( 1 + 2.96T + 41T^{2} \)
43 \( 1 + 6.74T + 43T^{2} \)
47 \( 1 + 7.14T + 47T^{2} \)
53 \( 1 + 5.09T + 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + 1.28T + 61T^{2} \)
67 \( 1 + 4.97T + 67T^{2} \)
71 \( 1 + 2.39T + 71T^{2} \)
73 \( 1 - 1.07T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 7.18T + 83T^{2} \)
89 \( 1 - 2.66T + 89T^{2} \)
97 \( 1 - 6.54T + 97T^{2} \)
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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

−7.17588485577835422762039573696, −6.70016712583976649751203898987, −6.13611972777944516330736426094, −5.07066320908183263622197102982, −4.61909918540774002538687773844, −3.91703428893970626510908824792, −3.27014596230494593627793434973, −2.65751116825617645174171007559, −1.64242832595835014483838859559, 0, 1.64242832595835014483838859559, 2.65751116825617645174171007559, 3.27014596230494593627793434973, 3.91703428893970626510908824792, 4.61909918540774002538687773844, 5.07066320908183263622197102982, 6.13611972777944516330736426094, 6.70016712583976649751203898987, 7.17588485577835422762039573696

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