Properties

Label 2-799-799.140-c0-0-4
Degree $2$
Conductor $799$
Sign $-0.615 - 0.788i$
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  − 2i·2-s + (1 − i)3-s − 3·4-s + (−2 − 2i)6-s + (−1 − i)7-s + 4i·8-s i·9-s + (−3 + 3i)12-s + (−2 + 2i)14-s + 5·16-s + 17-s − 2·18-s − 2·21-s + (4 + 4i)24-s i·25-s + ⋯
L(s)  = 1  − 2i·2-s + (1 − i)3-s − 3·4-s + (−2 − 2i)6-s + (−1 − i)7-s + 4i·8-s i·9-s + (−3 + 3i)12-s + (−2 + 2i)14-s + 5·16-s + 17-s − 2·18-s − 2·21-s + (4 + 4i)24-s i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.615 - 0.788i$
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: \((2,\ 799,\ (\ :0),\ -0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9542886114\)
\(L(\frac12)\) \(\approx\) \(0.9542886114\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
47 \( 1 + T \)
good2 \( 1 + 2iT - T^{2} \)
3 \( 1 + (-1 + i)T - iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + (1 + i)T + iT^{2} \)
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 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-1 - i)T + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.02162990233791658777688878230, −9.385843929804427785077256885783, −8.386354777601009821054263059587, −7.80485911257533043226855360494, −6.56318741009394828702748829611, −5.00742648746650945834870031272, −3.71067770523075155581245407326, −3.20984728687798741128797199671, −2.16326964126628773584340625955, −0.974320373461786274110774279262, 3.19389671564216599440055642828, 3.86665528610201218482511266896, 5.08622314534893812886407372604, 5.74635104893792175643103972437, 6.70849618064936116204528590558, 7.67952333185749838503850381614, 8.517647138644809170150742433872, 9.149004211195580521395692797172, 9.594040985757889731963605525407, 10.30813916166489070980551840416

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