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Properties

Label 2-799-799.610-c0-0-2
Degree $2$
Conductor $799$
Sign $-0.762 - 0.647i$
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 + (0.178 + 0.431i)3-s + 2.61i·4-s + (−0.339 + 0.820i)6-s + (−1.40 − 0.581i)7-s + (−2.17 + 2.17i)8-s + (0.552 − 0.552i)9-s + (−1.12 + 0.467i)12-s + (−1.10 − 2.67i)14-s − 3.23·16-s + (0.809 + 0.587i)17-s + 1.48·18-s − 0.710i·21-s + (−1.32 − 0.549i)24-s + (0.707 − 0.707i)25-s + ⋯
L(s)  = 1  + (1.34 + 1.34i)2-s + (0.178 + 0.431i)3-s + 2.61i·4-s + (−0.339 + 0.820i)6-s + (−1.40 − 0.581i)7-s + (−2.17 + 2.17i)8-s + (0.552 − 0.552i)9-s + (−1.12 + 0.467i)12-s + (−1.10 − 2.67i)14-s − 3.23·16-s + (0.809 + 0.587i)17-s + 1.48·18-s − 0.710i·21-s + (−1.32 − 0.549i)24-s + (0.707 − 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.762 - 0.647i$
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: \((2,\ 799,\ (\ :0),\ -0.762 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.791149651\)
\(L(\frac12)\) \(\approx\) \(1.791149651\)
\(L(1)\) 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 \( 1 + (-0.178 - 0.431i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (1.40 + 0.581i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.744 + 1.79i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (-0.144 - 0.0600i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.652 + 1.57i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.652 - 1.57i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - 1.90iT - T^{2} \)
97 \( 1 + (1.84 - 0.763i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.76351824808709325283824946112, −9.795047254867594086338887362558, −8.964745358251315492285173999695, −7.893382040596508706037848038904, −6.99652813505804169742255354611, −6.46335502430067702527733042729, −5.62257531571421170729725710615, −4.44843438381123805269516216009, −3.71907855731235172943011023353, −3.06285533267122985110171892950, 1.49259719282995548490097771660, 2.79135870140498464984674385396, 3.30827516886163669637572293290, 4.59435466099277358246153828577, 5.46698709075936681319563391281, 6.36882467384306850479505177358, 7.21656826530745335834952435561, 8.784605335479875652774326105159, 9.829195469556129938369046994322, 10.14977220069250446768653361901

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