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 − 0.915·4-s − 1.30·5-s + 1.04·6-s + 3.53·7-s − 3.03·8-s + 9-s − 1.35·10-s + 3.63·11-s − 0.915·12-s + 2.12·13-s + 3.67·14-s − 1.30·15-s − 1.33·16-s − 17-s + 1.04·18-s − 6.54·19-s + 1.19·20-s + 3.53·21-s + 3.78·22-s − 9.46·23-s − 3.03·24-s − 3.30·25-s + 2.21·26-s + 27-s − 3.23·28-s + ⋯
L(s)  = 1  + 0.736·2-s + 0.577·3-s − 0.457·4-s − 0.582·5-s + 0.425·6-s + 1.33·7-s − 1.07·8-s + 0.333·9-s − 0.428·10-s + 1.09·11-s − 0.264·12-s + 0.589·13-s + 0.983·14-s − 0.336·15-s − 0.332·16-s − 0.242·17-s + 0.245·18-s − 1.50·19-s + 0.266·20-s + 0.770·21-s + 0.806·22-s − 1.97·23-s − 0.619·24-s − 0.660·25-s + 0.434·26-s + 0.192·27-s − 0.611·28-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 \( 1 - 1.04T + 2T^{2} \)
5 \( 1 + 1.30T + 5T^{2} \)
7 \( 1 - 3.53T + 7T^{2} \)
11 \( 1 - 3.63T + 11T^{2} \)
13 \( 1 - 2.12T + 13T^{2} \)
19 \( 1 + 6.54T + 19T^{2} \)
23 \( 1 + 9.46T + 23T^{2} \)
29 \( 1 + 5.38T + 29T^{2} \)
31 \( 1 - 3.92T + 31T^{2} \)
37 \( 1 - 7.42T + 37T^{2} \)
41 \( 1 + 9.21T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + 3.29T + 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 + 15.2T + 59T^{2} \)
61 \( 1 + 0.736T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 7.08T + 71T^{2} \)
73 \( 1 - 8.51T + 73T^{2} \)
79 \( 1 - 5.94T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + 5.14T + 89T^{2} \)
97 \( 1 + 1.15T + 97T^{2} \)
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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

−7.84823673452257987107138499041, −6.46540134727024646805790006913, −6.25916272941436834047527318817, −5.10043645021134611103639793461, −4.51009542966259822729447899595, −3.88117185131382699290759744461, −3.59012537694997671072339124951, −2.20338594838448561386920777680, −1.53451937238242466821858401346, 0, 1.53451937238242466821858401346, 2.20338594838448561386920777680, 3.59012537694997671072339124951, 3.88117185131382699290759744461, 4.51009542966259822729447899595, 5.10043645021134611103639793461, 6.25916272941436834047527318817, 6.46540134727024646805790006913, 7.84823673452257987107138499041

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