Properties

Label 2-690-15.8-c1-0-38
Degree $2$
Conductor $690$
Sign $0.902 + 0.430i$
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 + (1.28 − 1.16i)3-s + 1.00i·4-s + (1.31 − 1.80i)5-s + (1.73 + 0.0807i)6-s + (1.43 − 1.43i)7-s + (−0.707 + 0.707i)8-s + (0.279 − 2.98i)9-s + (2.20 − 0.344i)10-s + 3.03i·11-s + (1.16 + 1.28i)12-s + (−1.30 − 1.30i)13-s + 2.03·14-s + (−0.417 − 3.85i)15-s − 1.00·16-s + (−0.0287 − 0.0287i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.739 − 0.673i)3-s + 0.500i·4-s + (0.589 − 0.807i)5-s + (0.706 + 0.0329i)6-s + (0.542 − 0.542i)7-s + (−0.250 + 0.250i)8-s + (0.0931 − 0.995i)9-s + (0.698 − 0.108i)10-s + 0.915i·11-s + (0.336 + 0.369i)12-s + (−0.361 − 0.361i)13-s + 0.542·14-s + (−0.107 − 0.994i)15-s − 0.250·16-s + (−0.00696 − 0.00696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.902 + 0.430i$
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: \((2,\ 690,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71257 - 0.614515i\)
\(L(\frac12)\) \(\approx\) \(2.71257 - 0.614515i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 1.16i)T \)
5 \( 1 + (-1.31 + 1.80i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-1.43 + 1.43i)T - 7iT^{2} \)
11 \( 1 - 3.03iT - 11T^{2} \)
13 \( 1 + (1.30 + 1.30i)T + 13iT^{2} \)
17 \( 1 + (0.0287 + 0.0287i)T + 17iT^{2} \)
19 \( 1 - 2.41iT - 19T^{2} \)
29 \( 1 + 8.32T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + (0.399 - 0.399i)T - 37iT^{2} \)
41 \( 1 - 9.45iT - 41T^{2} \)
43 \( 1 + (0.108 + 0.108i)T + 43iT^{2} \)
47 \( 1 + (0.341 + 0.341i)T + 47iT^{2} \)
53 \( 1 + (0.120 - 0.120i)T - 53iT^{2} \)
59 \( 1 + 2.10T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + (6.68 - 6.68i)T - 67iT^{2} \)
71 \( 1 - 5.69iT - 71T^{2} \)
73 \( 1 + (-6.43 - 6.43i)T + 73iT^{2} \)
79 \( 1 + 5.14iT - 79T^{2} \)
83 \( 1 + (-3.07 + 3.07i)T - 83iT^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (-7.38 + 7.38i)T - 97iT^{2} \)
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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

−10.11499045027582474694243201630, −9.455131794895933907789139175144, −8.379505406002066488234514065946, −7.81819060873604578597138477951, −6.96788360635786612671415523358, −5.98442967755276014841254940790, −4.89021215174935498624811156035, −4.05152308024362350062681984281, −2.57169924900171714277577208184, −1.36652455762975210090870236348, 2.00448984151526772661322939854, 2.83173680632191027215492479261, 3.79869101618625847825522601460, 4.99816483716673420111315361429, 5.77605776819768802730190540408, 6.94140891589510110176462442951, 8.110321722683488922572377228797, 9.056324991488749072153331364921, 9.700881288961277101551816605744, 10.64748153069618948068651236756

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