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  + 1.10·2-s + 3-s − 0.788·4-s + 2.79·5-s + 1.10·6-s − 1.33·7-s − 3.06·8-s + 9-s + 3.08·10-s − 0.596·11-s − 0.788·12-s + 1.71·13-s − 1.47·14-s + 2.79·15-s − 1.80·16-s − 17-s + 1.10·18-s − 2.45·19-s − 2.20·20-s − 1.33·21-s − 0.656·22-s + 0.281·23-s − 3.06·24-s + 2.83·25-s + 1.88·26-s + 27-s + 1.05·28-s + ⋯
L(s)  = 1  + 0.778·2-s + 0.577·3-s − 0.394·4-s + 1.25·5-s + 0.449·6-s − 0.505·7-s − 1.08·8-s + 0.333·9-s + 0.974·10-s − 0.179·11-s − 0.227·12-s + 0.474·13-s − 0.393·14-s + 0.722·15-s − 0.450·16-s − 0.242·17-s + 0.259·18-s − 0.562·19-s − 0.493·20-s − 0.292·21-s − 0.140·22-s + 0.0587·23-s − 0.626·24-s + 0.566·25-s + 0.369·26-s + 0.192·27-s + 0.199·28-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 \( 1 - 1.10T + 2T^{2} \)
5 \( 1 - 2.79T + 5T^{2} \)
7 \( 1 + 1.33T + 7T^{2} \)
11 \( 1 + 0.596T + 11T^{2} \)
13 \( 1 - 1.71T + 13T^{2} \)
19 \( 1 + 2.45T + 19T^{2} \)
23 \( 1 - 0.281T + 23T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 + 9.19T + 31T^{2} \)
37 \( 1 + 9.39T + 37T^{2} \)
41 \( 1 + 7.64T + 41T^{2} \)
43 \( 1 - 4.42T + 43T^{2} \)
47 \( 1 + 2.90T + 47T^{2} \)
53 \( 1 + 9.14T + 53T^{2} \)
59 \( 1 + 0.751T + 59T^{2} \)
61 \( 1 - 1.53T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 2.38T + 71T^{2} \)
73 \( 1 + 2.34T + 73T^{2} \)
79 \( 1 + 8.18T + 79T^{2} \)
83 \( 1 - 1.65T + 83T^{2} \)
89 \( 1 + 0.215T + 89T^{2} \)
97 \( 1 - 3.67T + 97T^{2} \)
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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

−7.35448845711518268117541712170, −6.54253424742924079130723521208, −6.04117924616884510379611799466, −5.33894902643317812656142773700, −4.76628170559568011393528818315, −3.71046939525949063231914543046, −3.33723129550006624481339655151, −2.32370409069405770534417730086, −1.61797003907202538300732764551, 0, 1.61797003907202538300732764551, 2.32370409069405770534417730086, 3.33723129550006624481339655151, 3.71046939525949063231914543046, 4.76628170559568011393528818315, 5.33894902643317812656142773700, 6.04117924616884510379611799466, 6.54253424742924079130723521208, 7.35448845711518268117541712170

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