↑

Properties

Label 2-799-799.610-c0-0-1
Degree $2$
Conductor $799$
Sign $-0.851 + 0.524i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 − 1.34i)2-s + (−0.652 − 1.57i)3-s + 2.61i·4-s + (−1.24 + 2.99i)6-s + (1.84 + 0.763i)7-s + (2.17 − 2.17i)8-s + (−1.34 + 1.34i)9-s + (4.12 − 1.70i)12-s + (−1.45 − 3.50i)14-s − 3.23·16-s + (0.809 − 0.587i)17-s + 3.62·18-s − 3.40i·21-s + (−4.84 − 2.00i)24-s + (0.707 − 0.707i)25-s + ⋯
L(s)  = 1  + (−1.34 − 1.34i)2-s + (−0.652 − 1.57i)3-s + 2.61i·4-s + (−1.24 + 2.99i)6-s + (1.84 + 0.763i)7-s + (2.17 − 2.17i)8-s + (−1.34 + 1.34i)9-s + (4.12 − 1.70i)12-s + (−1.45 − 3.50i)14-s − 3.23·16-s + (0.809 − 0.587i)17-s + 3.62·18-s − 3.40i·21-s + (−4.84 − 2.00i)24-s + (0.707 − 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (610, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ -0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4892778273\)
\(L(\frac12)\) \(\approx\) \(0.4892778273\)
\(L(1)\) 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 \( 1 + (0.652 + 1.57i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (-1.84 - 0.763i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.399 + 0.965i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
53 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
59 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
61 \( 1 + (1.20 + 0.497i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.178 - 0.431i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.178 + 0.431i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + 1.90iT - T^{2} \)
97 \( 1 + (-1.40 + 0.581i)T + (0.707 - 0.707i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.46426647464806852761771275632, −9.112366397121839277942475499320, −8.432569945083497528501987518454, −7.73293406678119248856901490041, −7.26488864299852338828188718038, −5.82668997362840164636937598674, −4.65601085692119525586761077705, −2.77320082433711147667319089125, −1.90185074092271568990911635741, −1.09419371937868918424388460561, 1.33350149001272049216378001712, 4.01975467616099495464128746581, 5.09812645347659382369107667599, 5.32939756610732164667534035618, 6.57535430819997860836480220253, 7.62256747061211535585541923560, 8.318719229979729375055034303209, 9.046744766239264754631733189418, 9.998050059243759222201349561948, 10.54095560913065629224813232050

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