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  − 2.66·2-s − 3-s + 5.11·4-s − 0.759·5-s + 2.66·6-s − 3.14·7-s − 8.31·8-s + 9-s + 2.02·10-s + 2.82·11-s − 5.11·12-s − 3.24·13-s + 8.39·14-s + 0.759·15-s + 11.9·16-s − 17-s − 2.66·18-s + 6.68·19-s − 3.88·20-s + 3.14·21-s − 7.54·22-s + 3.37·23-s + 8.31·24-s − 4.42·25-s + 8.66·26-s − 27-s − 16.1·28-s + ⋯
L(s)  = 1  − 1.88·2-s − 0.577·3-s + 2.55·4-s − 0.339·5-s + 1.08·6-s − 1.18·7-s − 2.93·8-s + 0.333·9-s + 0.640·10-s + 0.853·11-s − 1.47·12-s − 0.900·13-s + 2.24·14-s + 0.196·15-s + 2.98·16-s − 0.242·17-s − 0.628·18-s + 1.53·19-s − 0.869·20-s + 0.686·21-s − 1.60·22-s + 0.704·23-s + 1.69·24-s − 0.884·25-s + 1.69·26-s − 0.192·27-s − 3.04·28-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 \( 1 + 2.66T + 2T^{2} \)
5 \( 1 + 0.759T + 5T^{2} \)
7 \( 1 + 3.14T + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 3.24T + 13T^{2} \)
19 \( 1 - 6.68T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 - 2.88T + 31T^{2} \)
37 \( 1 + 4.84T + 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 + 7.84T + 43T^{2} \)
47 \( 1 + 8.89T + 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 + 2.68T + 59T^{2} \)
61 \( 1 - 0.513T + 61T^{2} \)
67 \( 1 - 0.811T + 67T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 - 9.87T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 2.15T + 83T^{2} \)
89 \( 1 - 1.54T + 89T^{2} \)
97 \( 1 + 1.93T + 97T^{2} \)
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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

−7.46166399441056459849595657409, −6.90860902150200136850386601363, −6.56276393920172807929245411811, −5.76017847860095030184713669854, −4.81043972487456471521736393308, −3.50722768487765199506577625065, −2.95147823654859037548608567369, −1.82830977266773974916285997054, −0.868185449589114862515433027515, 0, 0.868185449589114862515433027515, 1.82830977266773974916285997054, 2.95147823654859037548608567369, 3.50722768487765199506577625065, 4.81043972487456471521736393308, 5.76017847860095030184713669854, 6.56276393920172807929245411811, 6.90860902150200136850386601363, 7.46166399441056459849595657409

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