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Properties

Label 2-799-799.140-c0-0-2
Degree $2$
Conductor $799$
Sign $0.0345 + 0.999i$
Analytic cond. $0.398752$
Root an. cond. $0.631468$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618i·2-s + (−1.39 + 1.39i)3-s + 0.618·4-s + (0.863 + 0.863i)6-s + (−1.26 − 1.26i)7-s − i·8-s − 2.90i·9-s + (−0.863 + 0.863i)12-s + (−0.778 + 0.778i)14-s + (−0.809 − 0.587i)17-s − 1.79·18-s + 3.52·21-s + (1.39 + 1.39i)24-s − i·25-s + (2.65 + 2.65i)27-s + (−0.778 − 0.778i)28-s + ⋯
L(s)  = 1  − 0.618i·2-s + (−1.39 + 1.39i)3-s + 0.618·4-s + (0.863 + 0.863i)6-s + (−1.26 − 1.26i)7-s − i·8-s − 2.90i·9-s + (−0.863 + 0.863i)12-s + (−0.778 + 0.778i)14-s + (−0.809 − 0.587i)17-s − 1.79·18-s + 3.52·21-s + (1.39 + 1.39i)24-s − i·25-s + (2.65 + 2.65i)27-s + (−0.778 − 0.778i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0345 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0345 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.0345 + 0.999i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 0.0345 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5189298182\)
\(L(\frac12)\) \(\approx\) \(0.5189298182\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + T \)
good2 \( 1 + 0.618iT - T^{2} \)
3 \( 1 + (1.39 - 1.39i)T - iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + T^{2} \)
53 \( 1 - 1.90iT - T^{2} \)
59 \( 1 + 0.618iT - T^{2} \)
61 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.221 - 0.221i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.221 + 0.221i)T + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48200864939980629698108459808, −9.758183804580998022164819647660, −9.264538702161785770244449819318, −7.32599039108412008074659300530, −6.50083124556308574273904749538, −6.01307368929553089493759453885, −4.56047503026960774596711923914, −3.93264964012453810592051906430, −3.00094901969608208388685868842, −0.60800463655263510644104495712, 1.81084080736197655401449178659, 2.81346149624886806405363955457, 5.03508037943819813876962655512, 5.84162491915296000265316766265, 6.37722240273964005316283525427, 6.84297995113939007710331975539, 7.81756446824782878727755370523, 8.661325433397868480039381897762, 9.969055956842267580343363363345, 11.02954928689355618487886980649

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