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Properties

Label 2-690-15.8-c1-0-37
Degree $2$
Conductor $690$
Sign $0.614 + 0.789i$
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  + (0.707 + 0.707i)2-s + (0.987 − 1.42i)3-s + 1.00i·4-s + (−2.16 + 0.556i)5-s + (1.70 − 0.307i)6-s + (3.32 − 3.32i)7-s + (−0.707 + 0.707i)8-s + (−1.04 − 2.81i)9-s + (−1.92 − 1.13i)10-s − 1.84i·11-s + (1.42 + 0.987i)12-s + (−1.71 − 1.71i)13-s + 4.69·14-s + (−1.34 + 3.63i)15-s − 1.00·16-s + (0.948 + 0.948i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.570 − 0.821i)3-s + 0.500i·4-s + (−0.968 + 0.248i)5-s + (0.695 − 0.125i)6-s + (1.25 − 1.25i)7-s + (−0.250 + 0.250i)8-s + (−0.349 − 0.936i)9-s + (−0.608 − 0.359i)10-s − 0.557i·11-s + (0.410 + 0.285i)12-s + (−0.476 − 0.476i)13-s + 1.25·14-s + (−0.348 + 0.937i)15-s − 0.250·16-s + (0.229 + 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.614 + 0.789i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.614 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89429 - 0.925841i\)
\(L(\frac12)\) \(\approx\) \(1.89429 - 0.925841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.987 + 1.42i)T \)
5 \( 1 + (2.16 - 0.556i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-3.32 + 3.32i)T - 7iT^{2} \)
11 \( 1 + 1.84iT - 11T^{2} \)
13 \( 1 + (1.71 + 1.71i)T + 13iT^{2} \)
17 \( 1 + (-0.948 - 0.948i)T + 17iT^{2} \)
19 \( 1 + 1.25iT - 19T^{2} \)
29 \( 1 - 6.59T + 29T^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 + (-6.93 + 6.93i)T - 37iT^{2} \)
41 \( 1 - 0.505iT - 41T^{2} \)
43 \( 1 + (-6.74 - 6.74i)T + 43iT^{2} \)
47 \( 1 + (-7.17 - 7.17i)T + 47iT^{2} \)
53 \( 1 + (7.30 - 7.30i)T - 53iT^{2} \)
59 \( 1 + 7.06T + 59T^{2} \)
61 \( 1 - 3.82T + 61T^{2} \)
67 \( 1 + (2.57 - 2.57i)T - 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (3.30 + 3.30i)T + 73iT^{2} \)
79 \( 1 - 7.20iT - 79T^{2} \)
83 \( 1 + (11.2 - 11.2i)T - 83iT^{2} \)
89 \( 1 - 9.35T + 89T^{2} \)
97 \( 1 + (5.09 - 5.09i)T - 97iT^{2} \)
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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

−10.69953984275091281366564375251, −9.149185576012991136670673823758, −8.003084007772672767159158242707, −7.78977333418524077087387460029, −7.12440018163674647675026216019, −6.02536891534670544561590024927, −4.65746292245392703279241770332, −3.87462032395170272439983893427, −2.76679714014940634296174214567, −0.955175559233697625126231046215, 1.93918203429443793125451764573, 2.96577017842036745573030365295, 4.28167397639202716881629076885, 4.77488045217775873518742414980, 5.63640540583407121065329692507, 7.31371101757904287341322814700, 8.255836339962369137150194571861, 8.844961750080546111711175502631, 9.742006741266472676022165290033, 10.73155469364909607233619667672

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