Properties

Label 2-690-15.8-c1-0-11
Degree $2$
Conductor $690$
Sign $-0.179 - 0.983i$
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 + (−0.0456 + 1.73i)3-s + 1.00i·4-s + (1.54 + 1.62i)5-s + (1.25 − 1.19i)6-s + (0.528 − 0.528i)7-s + (0.707 − 0.707i)8-s + (−2.99 − 0.157i)9-s + (0.0565 − 2.23i)10-s + 4.60i·11-s + (−1.73 − 0.0456i)12-s + (4.73 + 4.73i)13-s − 0.746·14-s + (−2.87 + 2.59i)15-s − 1.00·16-s + (−3.32 − 3.32i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.0263 + 0.999i)3-s + 0.500i·4-s + (0.688 + 0.724i)5-s + (0.512 − 0.486i)6-s + (0.199 − 0.199i)7-s + (0.250 − 0.250i)8-s + (−0.998 − 0.0526i)9-s + (0.0178 − 0.706i)10-s + 1.38i·11-s + (−0.499 − 0.0131i)12-s + (1.31 + 1.31i)13-s − 0.199·14-s + (−0.742 + 0.669i)15-s − 0.250·16-s + (−0.807 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.179 - 0.983i$
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: \((2,\ 690,\ (\ :1/2),\ -0.179 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770088 + 0.923040i\)
\(L(\frac12)\) \(\approx\) \(0.770088 + 0.923040i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0456 - 1.73i)T \)
5 \( 1 + (-1.54 - 1.62i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-0.528 + 0.528i)T - 7iT^{2} \)
11 \( 1 - 4.60iT - 11T^{2} \)
13 \( 1 + (-4.73 - 4.73i)T + 13iT^{2} \)
17 \( 1 + (3.32 + 3.32i)T + 17iT^{2} \)
19 \( 1 + 5.21iT - 19T^{2} \)
29 \( 1 - 2.50T + 29T^{2} \)
31 \( 1 - 3.91T + 31T^{2} \)
37 \( 1 + (7.08 - 7.08i)T - 37iT^{2} \)
41 \( 1 - 4.23iT - 41T^{2} \)
43 \( 1 + (7.82 + 7.82i)T + 43iT^{2} \)
47 \( 1 + (2.42 + 2.42i)T + 47iT^{2} \)
53 \( 1 + (1.66 - 1.66i)T - 53iT^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 1.51T + 61T^{2} \)
67 \( 1 + (5.25 - 5.25i)T - 67iT^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (5.55 + 5.55i)T + 73iT^{2} \)
79 \( 1 - 6.67iT - 79T^{2} \)
83 \( 1 + (-9.68 + 9.68i)T - 83iT^{2} \)
89 \( 1 - 9.07T + 89T^{2} \)
97 \( 1 + (-5.69 + 5.69i)T - 97iT^{2} \)
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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

−10.53306767487820327094717261330, −9.930966342052937065084053361085, −9.190962218765451996133170038429, −8.563644110768652080477392905108, −7.04336880351946397575115815645, −6.50736548488773540944374194688, −4.97109208349191507961811644502, −4.20634216617305248971501729015, −2.96968904946992312839980087064, −1.86584763498710840710596211645, 0.77312771749141007857409333079, 1.86809207981868575421082800494, 3.45158525797000377861853751585, 5.28039484351515717613092263791, 6.00156817640891169165663133324, 6.41226317650264842139404432878, 7.996702413673325359682870778210, 8.394960790824936533840936330349, 8.881319809478980915820746085348, 10.29311673580073063319041160880

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