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Properties

Label 2-799-799.281-c0-0-1
Degree $2$
Conductor $799$
Sign $-0.110 + 0.993i$
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  + (−1.34 + 1.34i)2-s + (−1.79 − 0.744i)3-s − 2.61i·4-s + (3.41 − 1.41i)6-s + (−0.497 − 1.20i)7-s + (2.17 + 2.17i)8-s + (1.96 + 1.96i)9-s + (−1.94 + 4.70i)12-s + (2.28 + 0.945i)14-s − 3.23·16-s + (0.809 − 0.587i)17-s − 5.29·18-s + 2.52i·21-s + (−2.29 − 5.52i)24-s + (−0.707 − 0.707i)25-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)2-s + (−1.79 − 0.744i)3-s − 2.61i·4-s + (3.41 − 1.41i)6-s + (−0.497 − 1.20i)7-s + (2.17 + 2.17i)8-s + (1.96 + 1.96i)9-s + (−1.94 + 4.70i)12-s + (2.28 + 0.945i)14-s − 3.23·16-s + (0.809 − 0.587i)17-s − 5.29·18-s + 2.52i·21-s + (−2.29 − 5.52i)24-s + (−0.707 − 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.110 + 0.993i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ -0.110 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1500840517\)
\(L(\frac12)\) \(\approx\) \(0.1500840517\)
\(L(1)\) 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 \( 1 + iT \)
good2 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
3 \( 1 + (1.79 + 0.744i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.497 + 1.20i)T + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.431 + 0.178i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
59 \( 1 + (-0.437 - 0.437i)T + iT^{2} \)
61 \( 1 + (0.763 + 1.84i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.965 + 0.399i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.965 - 0.399i)T + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + 1.90iT - T^{2} \)
97 \( 1 + (0.0600 - 0.144i)T + (-0.707 - 0.707i)T^{2} \)
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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

−10.21185058585300800755513678911, −9.584114065715587147620555072778, −8.109563004768069807470082615562, −7.38978085549222278297698525628, −6.92362943470849987010976260463, −6.16810585498017789857709077496, −5.47813846953322668509285812130, −4.49071472176212154672165672325, −1.47059383884631236045086779005, −0.33836518353393647209040843793, 1.48238663923497214537060454852, 3.12260494017461346668307122768, 4.12782769776889355281137548818, 5.40576608198262380230051190683, 6.26883472649525349395970380522, 7.41276096851232150832535343069, 8.630632746563601170957524209486, 9.505808394417964924429625845604, 9.915995878999899261931353848984, 10.68322044600543759492919390727

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