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  + 1.04·2-s − 3-s − 0.901·4-s + 2.25·5-s − 1.04·6-s − 3.56·7-s − 3.04·8-s + 9-s + 2.36·10-s − 0.421·11-s + 0.901·12-s + 1.27·13-s − 3.73·14-s − 2.25·15-s − 1.38·16-s − 17-s + 1.04·18-s + 7.63·19-s − 2.03·20-s + 3.56·21-s − 0.441·22-s − 3.06·23-s + 3.04·24-s + 0.0948·25-s + 1.33·26-s − 27-s + 3.21·28-s + ⋯
L(s)  = 1  + 0.741·2-s − 0.577·3-s − 0.450·4-s + 1.00·5-s − 0.427·6-s − 1.34·7-s − 1.07·8-s + 0.333·9-s + 0.748·10-s − 0.127·11-s + 0.260·12-s + 0.352·13-s − 0.998·14-s − 0.582·15-s − 0.345·16-s − 0.242·17-s + 0.247·18-s + 1.75·19-s − 0.455·20-s + 0.778·21-s − 0.0941·22-s − 0.640·23-s + 0.620·24-s + 0.0189·25-s + 0.261·26-s − 0.192·27-s + 0.607·28-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 - 1.04T + 2T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 + 0.421T + 11T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
19 \( 1 - 7.63T + 19T^{2} \)
23 \( 1 + 3.06T + 23T^{2} \)
29 \( 1 - 6.05T + 29T^{2} \)
31 \( 1 - 0.390T + 31T^{2} \)
37 \( 1 + 8.69T + 37T^{2} \)
41 \( 1 + 5.09T + 41T^{2} \)
43 \( 1 - 4.23T + 43T^{2} \)
47 \( 1 + 4.45T + 47T^{2} \)
53 \( 1 - 5.96T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 3.31T + 61T^{2} \)
67 \( 1 + 9.42T + 67T^{2} \)
71 \( 1 + 3.49T + 71T^{2} \)
73 \( 1 + 2.53T + 73T^{2} \)
79 \( 1 - 3.30T + 79T^{2} \)
83 \( 1 - 7.63T + 83T^{2} \)
89 \( 1 + 6.18T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−7.10692131375270747473616966724, −6.57252694789037672684781158124, −5.89300401513794177939116911490, −5.50781047659943164272603495003, −4.85756758275427168805430845602, −3.85895307033183082570398064181, −3.27981458355845013863521528206, −2.46939440459064785503337548325, −1.16920194707153782200621743969, 0, 1.16920194707153782200621743969, 2.46939440459064785503337548325, 3.27981458355845013863521528206, 3.85895307033183082570398064181, 4.85756758275427168805430845602, 5.50781047659943164272603495003, 5.89300401513794177939116911490, 6.57252694789037672684781158124, 7.10692131375270747473616966724

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