Properties

Degree 2
Conductor $ 7 \cdot 859 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.02·2-s + 2.43·3-s + 2.09·4-s + 0.154·5-s − 4.92·6-s + 7-s − 0.185·8-s + 2.93·9-s − 0.312·10-s − 6.02·11-s + 5.09·12-s + 2.43·13-s − 2.02·14-s + 0.376·15-s − 3.80·16-s + 4.14·17-s − 5.94·18-s + 0.405·19-s + 0.323·20-s + 2.43·21-s + 12.1·22-s + 9.04·23-s − 0.451·24-s − 4.97·25-s − 4.93·26-s − 0.152·27-s + 2.09·28-s + ⋯
L(s)  = 1  − 1.43·2-s + 1.40·3-s + 1.04·4-s + 0.0691·5-s − 2.01·6-s + 0.377·7-s − 0.0654·8-s + 0.979·9-s − 0.0989·10-s − 1.81·11-s + 1.47·12-s + 0.676·13-s − 0.540·14-s + 0.0973·15-s − 0.952·16-s + 1.00·17-s − 1.40·18-s + 0.0931·19-s + 0.0723·20-s + 0.531·21-s + 2.59·22-s + 1.88·23-s − 0.0920·24-s − 0.995·25-s − 0.967·26-s − 0.0293·27-s + 0.395·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6013\)    =    \(7 \cdot 859\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6013} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6013,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.634959280\)
\(L(\frac12)\)  \(\approx\)  \(1.634959280\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{7,\;859\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;859\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 - T \)
859 \( 1 + T \)
good2 \( 1 + 2.02T + 2T^{2} \)
3 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 - 0.154T + 5T^{2} \)
11 \( 1 + 6.02T + 11T^{2} \)
13 \( 1 - 2.43T + 13T^{2} \)
17 \( 1 - 4.14T + 17T^{2} \)
19 \( 1 - 0.405T + 19T^{2} \)
23 \( 1 - 9.04T + 23T^{2} \)
29 \( 1 - 2.00T + 29T^{2} \)
31 \( 1 - 2.00T + 31T^{2} \)
37 \( 1 - 5.46T + 37T^{2} \)
41 \( 1 + 2.37T + 41T^{2} \)
43 \( 1 - 7.65T + 43T^{2} \)
47 \( 1 - 5.36T + 47T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 - 0.593T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + 3.94T + 67T^{2} \)
71 \( 1 - 1.58T + 71T^{2} \)
73 \( 1 + 5.90T + 73T^{2} \)
79 \( 1 + 4.92T + 79T^{2} \)
83 \( 1 + 4.58T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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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

−8.085639138768476165633798513530, −7.63730397934953648588365883098, −7.39420164587400488590778893710, −6.11979702852409487092505464078, −5.20844802729651927412571253076, −4.34420889241920903875566944916, −3.16146045745129031090249156380, −2.66862058599159747709333065933, −1.77738107982301317135260666293, −0.796087187997522618595648994764, 0.796087187997522618595648994764, 1.77738107982301317135260666293, 2.66862058599159747709333065933, 3.16146045745129031090249156380, 4.34420889241920903875566944916, 5.20844802729651927412571253076, 6.11979702852409487092505464078, 7.39420164587400488590778893710, 7.63730397934953648588365883098, 8.085639138768476165633798513530

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