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  − 2.38·2-s + 3-s + 3.67·4-s − 2.44·5-s − 2.38·6-s − 2.76·7-s − 3.99·8-s + 9-s + 5.83·10-s + 5.16·11-s + 3.67·12-s + 5.84·13-s + 6.60·14-s − 2.44·15-s + 2.17·16-s − 17-s − 2.38·18-s + 7.85·19-s − 9.00·20-s − 2.76·21-s − 12.2·22-s − 7.69·23-s − 3.99·24-s + 0.997·25-s − 13.9·26-s + 27-s − 10.1·28-s + ⋯
L(s)  = 1  − 1.68·2-s + 0.577·3-s + 1.83·4-s − 1.09·5-s − 0.972·6-s − 1.04·7-s − 1.41·8-s + 0.333·9-s + 1.84·10-s + 1.55·11-s + 1.06·12-s + 1.62·13-s + 1.76·14-s − 0.632·15-s + 0.542·16-s − 0.242·17-s − 0.561·18-s + 1.80·19-s − 2.01·20-s − 0.604·21-s − 2.62·22-s − 1.60·23-s − 0.816·24-s + 0.199·25-s − 2.73·26-s + 0.192·27-s − 1.92·28-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 \( 1 + 2.38T + 2T^{2} \)
5 \( 1 + 2.44T + 5T^{2} \)
7 \( 1 + 2.76T + 7T^{2} \)
11 \( 1 - 5.16T + 11T^{2} \)
13 \( 1 - 5.84T + 13T^{2} \)
19 \( 1 - 7.85T + 19T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 + 0.463T + 29T^{2} \)
31 \( 1 + 0.798T + 31T^{2} \)
37 \( 1 + 2.00T + 37T^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 + 7.76T + 43T^{2} \)
47 \( 1 - 2.81T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 4.14T + 59T^{2} \)
61 \( 1 - 0.116T + 61T^{2} \)
67 \( 1 - 1.94T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 4.88T + 79T^{2} \)
83 \( 1 + 9.14T + 83T^{2} \)
89 \( 1 + 2.81T + 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−7.66756024377514747476478144642, −7.03917389153215537056392644512, −6.52100130155398584194617770258, −5.82160282676428310053314412716, −4.27892298276076868302348421511, −3.54263301846782600012484194816, −3.24483624594152590433520977942, −1.79668610808435621681456221010, −1.11657940033719194465788730654, 0, 1.11657940033719194465788730654, 1.79668610808435621681456221010, 3.24483624594152590433520977942, 3.54263301846782600012484194816, 4.27892298276076868302348421511, 5.82160282676428310053314412716, 6.52100130155398584194617770258, 7.03917389153215537056392644512, 7.66756024377514747476478144642

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