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  + 2.52·2-s + 3-s + 4.39·4-s − 3.06·5-s + 2.52·6-s − 1.89·7-s + 6.05·8-s + 9-s − 7.74·10-s − 6.10·11-s + 4.39·12-s + 6.06·13-s − 4.78·14-s − 3.06·15-s + 6.52·16-s − 17-s + 2.52·18-s + 1.41·19-s − 13.4·20-s − 1.89·21-s − 15.4·22-s + 0.697·23-s + 6.05·24-s + 4.38·25-s + 15.3·26-s + 27-s − 8.32·28-s + ⋯
L(s)  = 1  + 1.78·2-s + 0.577·3-s + 2.19·4-s − 1.36·5-s + 1.03·6-s − 0.715·7-s + 2.14·8-s + 0.333·9-s − 2.44·10-s − 1.84·11-s + 1.26·12-s + 1.68·13-s − 1.27·14-s − 0.790·15-s + 1.63·16-s − 0.242·17-s + 0.596·18-s + 0.325·19-s − 3.01·20-s − 0.413·21-s − 3.29·22-s + 0.145·23-s + 1.23·24-s + 0.876·25-s + 3.00·26-s + 0.192·27-s − 1.57·28-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 \( 1 - 2.52T + 2T^{2} \)
5 \( 1 + 3.06T + 5T^{2} \)
7 \( 1 + 1.89T + 7T^{2} \)
11 \( 1 + 6.10T + 11T^{2} \)
13 \( 1 - 6.06T + 13T^{2} \)
19 \( 1 - 1.41T + 19T^{2} \)
23 \( 1 - 0.697T + 23T^{2} \)
29 \( 1 + 6.79T + 29T^{2} \)
31 \( 1 - 0.344T + 31T^{2} \)
37 \( 1 + 3.08T + 37T^{2} \)
41 \( 1 + 4.03T + 41T^{2} \)
43 \( 1 + 0.755T + 43T^{2} \)
47 \( 1 + 4.00T + 47T^{2} \)
53 \( 1 + 0.627T + 53T^{2} \)
59 \( 1 + 7.13T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 6.27T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 0.512T + 73T^{2} \)
79 \( 1 + 6.80T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 6.41T + 89T^{2} \)
97 \( 1 + 9.88T + 97T^{2} \)
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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

−7.34726421034283061564138592059, −6.73510921486994063871985117440, −5.92162407673459418666409915304, −5.27972107409217623824346312263, −4.49334024177748624235865560473, −3.80669393160378172630935977924, −3.27082599183941199650132536544, −2.87395247559088418854819902141, −1.71472671797351718229954201881, 0, 1.71472671797351718229954201881, 2.87395247559088418854819902141, 3.27082599183941199650132536544, 3.80669393160378172630935977924, 4.49334024177748624235865560473, 5.27972107409217623824346312263, 5.92162407673459418666409915304, 6.73510921486994063871985117440, 7.34726421034283061564138592059

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