Properties

Label 2-690-15.2-c1-0-34
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: $\chi_{690} (47, \cdot )$
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

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

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