Properties

Label 2-690-69.11-c1-0-12
Degree $2$
Conductor $690$
Sign $0.994 - 0.107i$
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 + (1.50 + 0.864i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.0845 − 1.72i)6-s + (0.920 − 0.797i)7-s + (0.989 − 0.142i)8-s + (1.50 + 2.59i)9-s + (0.755 + 0.654i)10-s + (−0.473 − 0.304i)11-s + (−1.40 + 1.00i)12-s + (1.86 − 2.15i)13-s + (−1.16 − 0.343i)14-s + (−1.68 − 0.406i)15-s + (−0.654 − 0.755i)16-s + (1.73 + 3.79i)17-s + ⋯
L(s)  = 1  + (−0.382 − 0.594i)2-s + (0.866 + 0.498i)3-s + (−0.207 + 0.454i)4-s + (−0.429 + 0.125i)5-s + (−0.0345 − 0.706i)6-s + (0.348 − 0.301i)7-s + (0.349 − 0.0503i)8-s + (0.502 + 0.864i)9-s + (0.238 + 0.207i)10-s + (−0.142 − 0.0918i)11-s + (−0.406 + 0.290i)12-s + (0.517 − 0.597i)13-s + (−0.312 − 0.0917i)14-s + (−0.434 − 0.104i)15-s + (−0.163 − 0.188i)16-s + (0.420 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.994 - 0.107i$
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: \((2,\ 690,\ (\ :1/2),\ 0.994 - 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64382 + 0.0882889i\)
\(L(\frac12)\) \(\approx\) \(1.64382 + 0.0882889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 - 0.864i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (4.34 + 2.02i)T \)
good7 \( 1 + (-0.920 + 0.797i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (0.473 + 0.304i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-1.86 + 2.15i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-1.73 - 3.79i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-6.59 - 3.01i)T + (12.4 + 14.3i)T^{2} \)
29 \( 1 + (-3.46 + 1.58i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.639 - 4.45i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-0.307 + 1.04i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (0.199 + 0.680i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (-12.6 - 1.81i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 - 3.93iT - 47T^{2} \)
53 \( 1 + (0.983 + 1.13i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (3.68 + 3.19i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (4.64 - 0.667i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (4.57 + 7.11i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (4.36 + 6.79i)T + (-29.4 + 64.5i)T^{2} \)
73 \( 1 + (-0.204 + 0.448i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (3.10 + 2.69i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (-4.05 - 1.18i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (0.605 - 4.20i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (3.38 + 11.5i)T + (-81.6 + 52.4i)T^{2} \)
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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

−10.46452565838734205643652722563, −9.734640246282036584908042214702, −8.768056292485814639277306169911, −7.917673750271796377064284634815, −7.60197070846507884307131776382, −5.93657209804403407871121119325, −4.61118742713036664348401523575, −3.69422272616350343544143033650, −2.90628096780983556499655673778, −1.39984683062917489210402874523, 1.10887155201363995676806840398, 2.60248164272358685472239593109, 3.88999108090369278096939708537, 5.07329207588937558538654442138, 6.20470003375215385673392403446, 7.34743215280004626723069618937, 7.67943183217813923645725558583, 8.717849534264938858990128933029, 9.270803887646397566616932214753, 10.11369952418525803677438724487

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