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  + 0.188·2-s − 3-s − 1.96·4-s + 3.42·5-s − 0.188·6-s + 1.78·7-s − 0.747·8-s + 9-s + 0.645·10-s + 3.14·11-s + 1.96·12-s − 5.35·13-s + 0.336·14-s − 3.42·15-s + 3.78·16-s − 17-s + 0.188·18-s − 0.832·19-s − 6.72·20-s − 1.78·21-s + 0.593·22-s − 4.73·23-s + 0.747·24-s + 6.72·25-s − 1.01·26-s − 27-s − 3.50·28-s + ⋯
L(s)  = 1  + 0.133·2-s − 0.577·3-s − 0.982·4-s + 1.53·5-s − 0.0769·6-s + 0.674·7-s − 0.264·8-s + 0.333·9-s + 0.204·10-s + 0.949·11-s + 0.567·12-s − 1.48·13-s + 0.0898·14-s − 0.884·15-s + 0.947·16-s − 0.242·17-s + 0.0444·18-s − 0.190·19-s − 1.50·20-s − 0.389·21-s + 0.126·22-s − 0.986·23-s + 0.152·24-s + 1.34·25-s − 0.198·26-s − 0.192·27-s − 0.662·28-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 \( 1 - 0.188T + 2T^{2} \)
5 \( 1 - 3.42T + 5T^{2} \)
7 \( 1 - 1.78T + 7T^{2} \)
11 \( 1 - 3.14T + 11T^{2} \)
13 \( 1 + 5.35T + 13T^{2} \)
19 \( 1 + 0.832T + 19T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 - 5.10T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 + 6.02T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 - 8.72T + 53T^{2} \)
59 \( 1 - 4.56T + 59T^{2} \)
61 \( 1 + 1.80T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 3.53T + 71T^{2} \)
73 \( 1 + 9.39T + 73T^{2} \)
79 \( 1 + 6.65T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 - 6.57T + 97T^{2} \)
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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

−7.36802203062136121997665463609, −6.60913806566043783578093913467, −5.99516717417843498520475295634, −5.32018505271460631699131751424, −4.77651962748883851525719900238, −4.26283582036451084430019169691, −3.08223430931047752956731901942, −1.99502219373457835832504536010, −1.36552451734416153228407689814, 0, 1.36552451734416153228407689814, 1.99502219373457835832504536010, 3.08223430931047752956731901942, 4.26283582036451084430019169691, 4.77651962748883851525719900238, 5.32018505271460631699131751424, 5.99516717417843498520475295634, 6.60913806566043783578093913467, 7.36802203062136121997665463609

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