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Properties

Label 2-690-345.344-c1-0-43
Degree $2$
Conductor $690$
Sign $-0.626 + 0.779i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.571 + 1.63i)3-s + 4-s + (−2.12 − 0.689i)5-s + (−0.571 + 1.63i)6-s − 1.28·7-s + 8-s + (−2.34 − 1.86i)9-s + (−2.12 − 0.689i)10-s − 4.52·11-s + (−0.571 + 1.63i)12-s − 3.04i·13-s − 1.28·14-s + (2.34 − 3.08i)15-s + 16-s − 5.73i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.330 + 0.943i)3-s + 0.5·4-s + (−0.951 − 0.308i)5-s + (−0.233 + 0.667i)6-s − 0.487·7-s + 0.353·8-s + (−0.781 − 0.623i)9-s + (−0.672 − 0.217i)10-s − 1.36·11-s + (−0.165 + 0.471i)12-s − 0.843i·13-s − 0.344·14-s + (0.605 − 0.796i)15-s + 0.250·16-s − 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.626 + 0.779i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.626 + 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126960 - 0.265045i\)
\(L(\frac12)\) \(\approx\) \(0.126960 - 0.265045i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.571 - 1.63i)T \)
5 \( 1 + (2.12 + 0.689i)T \)
23 \( 1 + (4.79 + 0.132i)T \)
good7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 + 4.52T + 11T^{2} \)
13 \( 1 + 3.04iT - 13T^{2} \)
17 \( 1 + 5.73iT - 17T^{2} \)
19 \( 1 - 3.52iT - 19T^{2} \)
29 \( 1 + 3.03iT - 29T^{2} \)
31 \( 1 + 7.30T + 31T^{2} \)
37 \( 1 - 8.69T + 37T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 + 8.85T + 43T^{2} \)
47 \( 1 + 3.48T + 47T^{2} \)
53 \( 1 + 4.54iT - 53T^{2} \)
59 \( 1 + 12.7iT - 59T^{2} \)
61 \( 1 - 3.04iT - 61T^{2} \)
67 \( 1 + 3.98T + 67T^{2} \)
71 \( 1 - 5.30iT - 71T^{2} \)
73 \( 1 + 9.43iT - 73T^{2} \)
79 \( 1 - 16.0iT - 79T^{2} \)
83 \( 1 + 8.09iT - 83T^{2} \)
89 \( 1 + 6.63T + 89T^{2} \)
97 \( 1 + 0.0268T + 97T^{2} \)
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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

−10.17356285233114527722811365808, −9.626405485811967403749656000946, −8.207455096976533317241790214302, −7.67796753307667480870531382491, −6.32120599763338714512450622069, −5.32685865611521414766637810313, −4.72162862350203429822574320462, −3.60322612293362933121710504787, −2.85738647096754909888018464216, −0.11792702045958484868943438385, 2.03662735731724732026012957681, 3.18135113610831563772070478327, 4.31319833965110758399788669946, 5.47983882988932411921994303738, 6.37697608386640137577184064997, 7.20014047459184929669740022360, 7.86012355293524992428175525889, 8.764883379196638281644343330446, 10.33943654626192179619624703026, 10.98639353827496680820798400070

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