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Properties

Label 2-799-799.281-c0-0-2
Degree $2$
Conductor $799$
Sign $0.911 + 0.412i$
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  + (0.831 − 0.831i)2-s + (−0.144 − 0.0600i)3-s − 0.381i·4-s + (−0.170 + 0.0705i)6-s + (0.744 + 1.79i)7-s + (0.513 + 0.513i)8-s + (−0.689 − 0.689i)9-s + (−0.0229 + 0.0553i)12-s + (2.11 + 0.874i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s − 1.14·18-s − 0.305i·21-s + (−0.0436 − 0.105i)24-s + (−0.707 − 0.707i)25-s + ⋯
L(s)  = 1  + (0.831 − 0.831i)2-s + (−0.144 − 0.0600i)3-s − 0.381i·4-s + (−0.170 + 0.0705i)6-s + (0.744 + 1.79i)7-s + (0.513 + 0.513i)8-s + (−0.689 − 0.689i)9-s + (−0.0229 + 0.0553i)12-s + (2.11 + 0.874i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s − 1.14·18-s − 0.305i·21-s + (−0.0436 − 0.105i)24-s + (−0.707 − 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.911 + 0.412i$
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: \((2,\ 799,\ (\ :0),\ 0.911 + 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.467690454\)
\(L(\frac12)\) \(\approx\) \(1.467690454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + iT \)
good2 \( 1 + (-0.831 + 0.831i)T - iT^{2} \)
3 \( 1 + (0.144 + 0.0600i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.744 - 1.79i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (1.84 + 0.763i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
59 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
61 \( 1 + (-0.652 - 1.57i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.20 - 0.497i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1.20 + 0.497i)T + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 - 1.17iT - T^{2} \)
97 \( 1 + (0.399 - 0.965i)T + (-0.707 - 0.707i)T^{2} \)
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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

−10.83878884212750924866212340532, −9.579304294547858737368971953722, −8.707786781304350002030698189121, −8.136015347281656707824731521041, −6.74048579145474274465647678368, −5.48014779234737893977896054464, −5.22770284877695370330538520179, −3.88564385904656786407428291302, −2.76180661304722172829474014509, −2.03377961245795082116856778826, 1.55137492389192093949885263424, 3.58089174005224129702127668263, 4.40389014307780480914153795140, 5.14095174726300845463300754406, 6.08268329481494116685183119599, 7.06156797750850884516471775523, 7.68358171896417580852955392463, 8.456788595165184040628255638939, 9.932702572448796884036303536248, 10.69560395836249096797765019838

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