Properties

Label 2-690-115.68-c1-0-18
Degree $2$
Conductor $690$
Sign $-0.663 + 0.748i$
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 + (0.707 + 0.707i)3-s − 1.00i·4-s + (−2.19 + 0.408i)5-s + 1.00·6-s + (−1.64 − 1.64i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (−1.26 + 1.84i)10-s − 6.11i·11-s + (0.707 − 0.707i)12-s + (−0.241 − 0.241i)13-s − 2.32·14-s + (−1.84 − 1.26i)15-s − 1.00·16-s + (−0.327 − 0.327i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (−0.983 + 0.182i)5-s + 0.408·6-s + (−0.622 − 0.622i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.400 + 0.582i)10-s − 1.84i·11-s + (0.204 − 0.204i)12-s + (−0.0669 − 0.0669i)13-s − 0.622·14-s + (−0.475 − 0.326i)15-s − 0.250·16-s + (−0.0793 − 0.0793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.663 + 0.748i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507661 - 1.12830i\)
\(L(\frac12)\) \(\approx\) \(0.507661 - 1.12830i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.19 - 0.408i)T \)
23 \( 1 + (4.49 + 1.66i)T \)
good7 \( 1 + (1.64 + 1.64i)T + 7iT^{2} \)
11 \( 1 + 6.11iT - 11T^{2} \)
13 \( 1 + (0.241 + 0.241i)T + 13iT^{2} \)
17 \( 1 + (0.327 + 0.327i)T + 17iT^{2} \)
19 \( 1 - 0.759T + 19T^{2} \)
29 \( 1 + 7.09iT - 29T^{2} \)
31 \( 1 + 8.30T + 31T^{2} \)
37 \( 1 + (-1.92 - 1.92i)T + 37iT^{2} \)
41 \( 1 - 6.75T + 41T^{2} \)
43 \( 1 + (-0.144 + 0.144i)T - 43iT^{2} \)
47 \( 1 + (-1.46 + 1.46i)T - 47iT^{2} \)
53 \( 1 + (1.45 - 1.45i)T - 53iT^{2} \)
59 \( 1 + 1.68iT - 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 + (-3.13 - 3.13i)T + 67iT^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + (-6.62 - 6.62i)T + 73iT^{2} \)
79 \( 1 - 3.83T + 79T^{2} \)
83 \( 1 + (-2.14 + 2.14i)T - 83iT^{2} \)
89 \( 1 + 4.64T + 89T^{2} \)
97 \( 1 + (-11.2 - 11.2i)T + 97iT^{2} \)
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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

−10.36526580875323668987051597773, −9.407283276079139660153430724705, −8.438958313741886667206706915601, −7.65858653774997674825348922232, −6.49344811026557695266231034983, −5.52999774214330803916303606696, −4.11401430325786021899749762812, −3.65392995381710791062698741328, −2.70735771761282832705635109114, −0.51248249263051990343473445880, 2.07420723994742070417960407652, 3.37709474395665474510234880217, 4.30905772401216172603670158998, 5.32558038856729075752045360188, 6.54497543297374653060658933497, 7.33940395194524794709005735195, 7.87081207724180930668382598334, 9.019055138543208719121579411618, 9.597747817742710189607202924886, 10.93943376166894006303738617833

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