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Properties

Label 2-799-799.563-c0-0-3
Degree $2$
Conductor $799$
Sign $-0.494 + 0.868i$
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 + (1.20 − 0.497i)3-s + 0.381i·4-s + (−1.41 − 0.584i)6-s + (0.399 − 0.965i)7-s + (−0.513 + 0.513i)8-s + (0.485 − 0.485i)9-s + (0.189 + 0.458i)12-s + (−1.13 + 0.470i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s − 0.807·18-s − 1.35i·21-s + (−0.361 + 0.871i)24-s + (−0.707 + 0.707i)25-s + ⋯
L(s)  = 1  + (−0.831 − 0.831i)2-s + (1.20 − 0.497i)3-s + 0.381i·4-s + (−1.41 − 0.584i)6-s + (0.399 − 0.965i)7-s + (−0.513 + 0.513i)8-s + (0.485 − 0.485i)9-s + (0.189 + 0.458i)12-s + (−1.13 + 0.470i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s − 0.807·18-s − 1.35i·21-s + (−0.361 + 0.871i)24-s + (−0.707 + 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.494 + 0.868i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ -0.494 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9183235326\)
\(L(\frac12)\) \(\approx\) \(0.9183235326\)
\(L(1)\) 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 \( 1 + (-1.20 + 0.497i)T + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.399 + 0.965i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + (-1.40 + 0.581i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
53 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
59 \( 1 + (1.14 - 1.14i)T - iT^{2} \)
61 \( 1 + (0.178 - 0.431i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.144 - 0.0600i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.144 + 0.0600i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - 1.17iT - T^{2} \)
97 \( 1 + (0.744 + 1.79i)T + (-0.707 + 0.707i)T^{2} \)
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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

−10.04463652655954378794438625265, −9.311030322290994571514492987299, −8.704628026084515623626884273777, −7.70759010774840909745966918508, −7.34273531821674335096102022311, −5.86931289490206509448268439082, −4.46198662819902171694136371715, −3.22536410505826185438685865263, −2.32389001212998377344039451891, −1.22427847911943091610002951886, 2.15295545342666336722301994920, 3.27778941583657372297062469955, 4.32603634066652488165939285173, 5.73639637874246372396513966827, 6.55606143307828153361626014361, 7.82568541564049801503619116110, 8.264374332494591743899359536147, 8.868953056670942211435312533808, 9.538573716156041720598187770518, 10.25610175688448448646553069165

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