Properties

Label 2-690-69.11-c1-0-27
Degree $2$
Conductor $690$
Sign $0.964 + 0.263i$
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  + (0.540 + 0.841i)2-s + (1.37 − 1.05i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (1.63 + 0.583i)6-s + (2.43 − 2.10i)7-s + (−0.989 + 0.142i)8-s + (0.767 − 2.90i)9-s + (−0.755 − 0.654i)10-s + (−1.55 − 1.00i)11-s + (0.390 + 1.68i)12-s + (2.29 − 2.64i)13-s + (3.08 + 0.906i)14-s + (−1.01 + 1.40i)15-s + (−0.654 − 0.755i)16-s + (−1.24 − 2.72i)17-s + ⋯
L(s)  = 1  + (0.382 + 0.594i)2-s + (0.792 − 0.609i)3-s + (−0.207 + 0.454i)4-s + (−0.429 + 0.125i)5-s + (0.665 + 0.238i)6-s + (0.919 − 0.796i)7-s + (−0.349 + 0.0503i)8-s + (0.255 − 0.966i)9-s + (−0.238 − 0.207i)10-s + (−0.469 − 0.301i)11-s + (0.112 + 0.487i)12-s + (0.635 − 0.733i)13-s + (0.825 + 0.242i)14-s + (−0.263 + 0.361i)15-s + (−0.163 − 0.188i)16-s + (−0.301 − 0.660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33493 - 0.313716i\)
\(L(\frac12)\) \(\approx\) \(2.33493 - 0.313716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.540 - 0.841i)T \)
3 \( 1 + (-1.37 + 1.05i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (1.94 - 4.38i)T \)
good7 \( 1 + (-2.43 + 2.10i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (1.55 + 1.00i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-2.29 + 2.64i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (1.24 + 2.72i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-5.86 - 2.68i)T + (12.4 + 14.3i)T^{2} \)
29 \( 1 + (-2.52 + 1.15i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.0402 - 0.280i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (0.359 - 1.22i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-3.28 - 11.1i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (4.25 + 0.611i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 5.38iT - 47T^{2} \)
53 \( 1 + (-4.11 - 4.74i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-3.68 - 3.19i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (8.76 - 1.25i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (3.59 + 5.58i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (2.11 + 3.28i)T + (-29.4 + 64.5i)T^{2} \)
73 \( 1 + (-2.20 + 4.82i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-4.05 - 3.51i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (11.1 + 3.28i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.253 + 1.76i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-2.61 - 8.89i)T + (-81.6 + 52.4i)T^{2} \)
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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

−10.42470577099747007984092113975, −9.355327932776100261166123290800, −8.205915856299395944503738866306, −7.82694232304042918943061671211, −7.20930322187177368763377228702, −6.05912500049330218438876006907, −4.95243089838413277788970829090, −3.80590788718097309615883979332, −2.95695646435497542769012153671, −1.17277563250191458234554786248, 1.78523431336258709114414491692, 2.80415983213966096419564316602, 4.00189395594770263014663633980, 4.75736610715912549315592975444, 5.62294815192983423484033360513, 7.14047791668411200193962351402, 8.289201088516855418711688323274, 8.748608188858488892802226937570, 9.641449244348324655165447354674, 10.63119081069270060021764642965

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