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  − 2.28·2-s − 3-s + 3.21·4-s + 1.64·5-s + 2.28·6-s + 3.93·7-s − 2.77·8-s + 9-s − 3.75·10-s − 0.343·11-s − 3.21·12-s + 0.744·13-s − 8.98·14-s − 1.64·15-s − 0.0902·16-s − 17-s − 2.28·18-s − 7.57·19-s + 5.28·20-s − 3.93·21-s + 0.784·22-s − 3.77·23-s + 2.77·24-s − 2.29·25-s − 1.70·26-s − 27-s + 12.6·28-s + ⋯
L(s)  = 1  − 1.61·2-s − 0.577·3-s + 1.60·4-s + 0.734·5-s + 0.932·6-s + 1.48·7-s − 0.981·8-s + 0.333·9-s − 1.18·10-s − 0.103·11-s − 0.928·12-s + 0.206·13-s − 2.40·14-s − 0.424·15-s − 0.0225·16-s − 0.242·17-s − 0.538·18-s − 1.73·19-s + 1.18·20-s − 0.858·21-s + 0.167·22-s − 0.787·23-s + 0.566·24-s − 0.459·25-s − 0.333·26-s − 0.192·27-s + 2.39·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

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 \( 1 + 2.28T + 2T^{2} \)
5 \( 1 - 1.64T + 5T^{2} \)
7 \( 1 - 3.93T + 7T^{2} \)
11 \( 1 + 0.343T + 11T^{2} \)
13 \( 1 - 0.744T + 13T^{2} \)
19 \( 1 + 7.57T + 19T^{2} \)
23 \( 1 + 3.77T + 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 + 4.98T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 + 2.83T + 41T^{2} \)
43 \( 1 + 0.185T + 43T^{2} \)
47 \( 1 - 8.35T + 47T^{2} \)
53 \( 1 - 8.10T + 53T^{2} \)
59 \( 1 + 7.50T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 - 2.48T + 67T^{2} \)
71 \( 1 + 1.23T + 71T^{2} \)
73 \( 1 + 9.29T + 73T^{2} \)
79 \( 1 - 9.60T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + 3.05T + 89T^{2} \)
97 \( 1 + 6.82T + 97T^{2} \)
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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

−7.83511557860896591292948368370, −6.92192368168901449044645374031, −6.25574588873903996734353784732, −5.65229416526977884067217638415, −4.69993993396808027217479635723, −4.11311319224142896742995001935, −2.42354946533342827576214980346, −1.90736662720583020937389419265, −1.21016372259396342102788555266, 0, 1.21016372259396342102788555266, 1.90736662720583020937389419265, 2.42354946533342827576214980346, 4.11311319224142896742995001935, 4.69993993396808027217479635723, 5.65229416526977884067217638415, 6.25574588873903996734353784732, 6.92192368168901449044645374031, 7.83511557860896591292948368370

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