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  − 0.980·2-s − 3-s − 1.03·4-s + 1.00·5-s + 0.980·6-s − 3.53·7-s + 2.97·8-s + 9-s − 0.985·10-s + 6.32·11-s + 1.03·12-s + 3.86·13-s + 3.46·14-s − 1.00·15-s − 0.846·16-s − 17-s − 0.980·18-s − 2.59·19-s − 1.04·20-s + 3.53·21-s − 6.20·22-s + 6.45·23-s − 2.97·24-s − 3.99·25-s − 3.79·26-s − 27-s + 3.66·28-s + ⋯
L(s)  = 1  − 0.693·2-s − 0.577·3-s − 0.518·4-s + 0.449·5-s + 0.400·6-s − 1.33·7-s + 1.05·8-s + 0.333·9-s − 0.311·10-s + 1.90·11-s + 0.299·12-s + 1.07·13-s + 0.925·14-s − 0.259·15-s − 0.211·16-s − 0.242·17-s − 0.231·18-s − 0.595·19-s − 0.233·20-s + 0.770·21-s − 1.32·22-s + 1.34·23-s − 0.608·24-s − 0.798·25-s − 0.743·26-s − 0.192·27-s + 0.692·28-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 \( 1 + 0.980T + 2T^{2} \)
5 \( 1 - 1.00T + 5T^{2} \)
7 \( 1 + 3.53T + 7T^{2} \)
11 \( 1 - 6.32T + 11T^{2} \)
13 \( 1 - 3.86T + 13T^{2} \)
19 \( 1 + 2.59T + 19T^{2} \)
23 \( 1 - 6.45T + 23T^{2} \)
29 \( 1 + 7.53T + 29T^{2} \)
31 \( 1 + 4.09T + 31T^{2} \)
37 \( 1 - 6.50T + 37T^{2} \)
41 \( 1 + 9.61T + 41T^{2} \)
43 \( 1 - 7.33T + 43T^{2} \)
47 \( 1 + 4.71T + 47T^{2} \)
53 \( 1 - 2.16T + 53T^{2} \)
59 \( 1 - 3.64T + 59T^{2} \)
61 \( 1 - 7.27T + 61T^{2} \)
67 \( 1 + 6.61T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 8.05T + 83T^{2} \)
89 \( 1 - 2.16T + 89T^{2} \)
97 \( 1 - 3.94T + 97T^{2} \)
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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

−7.21831367470660528635650406546, −6.92772607562575534266973523894, −5.98948784784810188092814775855, −5.80518329661298004166481172116, −4.51803364576448047662310594436, −3.92404613644397706405324207769, −3.30700127716509272619982167035, −1.79937962502836335318712379453, −1.09545576732183928983972940713, 0, 1.09545576732183928983972940713, 1.79937962502836335318712379453, 3.30700127716509272619982167035, 3.92404613644397706405324207769, 4.51803364576448047662310594436, 5.80518329661298004166481172116, 5.98948784784810188092814775855, 6.92772607562575534266973523894, 7.21831367470660528635650406546

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