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  + 0.498·2-s − 3-s − 1.75·4-s + 0.622·5-s − 0.498·6-s + 0.110·7-s − 1.87·8-s + 9-s + 0.310·10-s + 0.189·11-s + 1.75·12-s + 6.61·13-s + 0.0548·14-s − 0.622·15-s + 2.56·16-s − 17-s + 0.498·18-s − 0.0616·19-s − 1.09·20-s − 0.110·21-s + 0.0947·22-s − 5.46·23-s + 1.87·24-s − 4.61·25-s + 3.30·26-s − 27-s − 0.192·28-s + ⋯
L(s)  = 1  + 0.352·2-s − 0.577·3-s − 0.875·4-s + 0.278·5-s − 0.203·6-s + 0.0415·7-s − 0.661·8-s + 0.333·9-s + 0.0982·10-s + 0.0572·11-s + 0.505·12-s + 1.83·13-s + 0.0146·14-s − 0.160·15-s + 0.642·16-s − 0.242·17-s + 0.117·18-s − 0.0141·19-s − 0.243·20-s − 0.0240·21-s + 0.0201·22-s − 1.14·23-s + 0.382·24-s − 0.922·25-s + 0.647·26-s − 0.192·27-s − 0.0364·28-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 \( 1 - 0.498T + 2T^{2} \)
5 \( 1 - 0.622T + 5T^{2} \)
7 \( 1 - 0.110T + 7T^{2} \)
11 \( 1 - 0.189T + 11T^{2} \)
13 \( 1 - 6.61T + 13T^{2} \)
19 \( 1 + 0.0616T + 19T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + 4.08T + 31T^{2} \)
37 \( 1 - 7.61T + 37T^{2} \)
41 \( 1 + 8.53T + 41T^{2} \)
43 \( 1 + 1.22T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + 0.0568T + 53T^{2} \)
59 \( 1 - 3.88T + 59T^{2} \)
61 \( 1 - 3.47T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + 6.33T + 73T^{2} \)
79 \( 1 + 2.39T + 79T^{2} \)
83 \( 1 - 17.9T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 9.70T + 97T^{2} \)
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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

−7.57029367367483568582152432692, −6.43691612081093638026216001262, −5.99005226092682241596752060050, −5.52212975777940813730918415678, −4.66759875437916479090641008567, −3.83899829606638902072553781392, −3.56327881395926606404629966050, −2.12102829379722973887988940137, −1.16459774122004603678270100432, 0, 1.16459774122004603678270100432, 2.12102829379722973887988940137, 3.56327881395926606404629966050, 3.83899829606638902072553781392, 4.66759875437916479090641008567, 5.52212975777940813730918415678, 5.99005226092682241596752060050, 6.43691612081093638026216001262, 7.57029367367483568582152432692

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