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  + 2.37·2-s − 3-s + 3.63·4-s − 3.39·5-s − 2.37·6-s − 2.72·7-s + 3.87·8-s + 9-s − 8.04·10-s + 0.191·11-s − 3.63·12-s + 3.63·13-s − 6.46·14-s + 3.39·15-s + 1.92·16-s − 17-s + 2.37·18-s + 6.06·19-s − 12.3·20-s + 2.72·21-s + 0.455·22-s + 7.64·23-s − 3.87·24-s + 6.49·25-s + 8.61·26-s − 27-s − 9.89·28-s + ⋯
L(s)  = 1  + 1.67·2-s − 0.577·3-s + 1.81·4-s − 1.51·5-s − 0.968·6-s − 1.02·7-s + 1.36·8-s + 0.333·9-s − 2.54·10-s + 0.0578·11-s − 1.04·12-s + 1.00·13-s − 1.72·14-s + 0.875·15-s + 0.481·16-s − 0.242·17-s + 0.559·18-s + 1.39·19-s − 2.75·20-s + 0.594·21-s + 0.0970·22-s + 1.59·23-s − 0.790·24-s + 1.29·25-s + 1.69·26-s − 0.192·27-s − 1.87·28-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 \( 1 - 2.37T + 2T^{2} \)
5 \( 1 + 3.39T + 5T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
11 \( 1 - 0.191T + 11T^{2} \)
13 \( 1 - 3.63T + 13T^{2} \)
19 \( 1 - 6.06T + 19T^{2} \)
23 \( 1 - 7.64T + 23T^{2} \)
29 \( 1 + 5.48T + 29T^{2} \)
31 \( 1 + 6.85T + 31T^{2} \)
37 \( 1 - 2.95T + 37T^{2} \)
41 \( 1 - 5.70T + 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 - 2.64T + 47T^{2} \)
53 \( 1 + 5.38T + 53T^{2} \)
59 \( 1 + 7.04T + 59T^{2} \)
61 \( 1 + 1.28T + 61T^{2} \)
67 \( 1 + 8.81T + 67T^{2} \)
71 \( 1 + 6.29T + 71T^{2} \)
73 \( 1 + 7.91T + 73T^{2} \)
79 \( 1 + 5.71T + 79T^{2} \)
83 \( 1 + 0.273T + 83T^{2} \)
89 \( 1 - 8.77T + 89T^{2} \)
97 \( 1 - 3.56T + 97T^{2} \)
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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

−7.36786792240977481887655299716, −6.59784940071398818846369578017, −5.97903873954848889089006105188, −5.32361559639611559793409414533, −4.58421411917827919464853310692, −3.88235605918528126482582817051, −3.38897432784492683025780225483, −2.87909900689215997853149673030, −1.29520717741806381173847091008, 0, 1.29520717741806381173847091008, 2.87909900689215997853149673030, 3.38897432784492683025780225483, 3.88235605918528126482582817051, 4.58421411917827919464853310692, 5.32361559639611559793409414533, 5.97903873954848889089006105188, 6.59784940071398818846369578017, 7.36786792240977481887655299716

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