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Properties

Label 2-690-69.68-c1-0-13
Degree $2$
Conductor $690$
Sign $0.989 - 0.144i$
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  + i·2-s + (−1.30 + 1.14i)3-s − 4-s − 5-s + (−1.14 − 1.30i)6-s − 1.02i·7-s − i·8-s + (0.384 − 2.97i)9-s − i·10-s + 0.336·11-s + (1.30 − 1.14i)12-s − 4.92·13-s + 1.02·14-s + (1.30 − 1.14i)15-s + 16-s + 7.57·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.751 + 0.660i)3-s − 0.5·4-s − 0.447·5-s + (−0.466 − 0.531i)6-s − 0.385i·7-s − 0.353i·8-s + (0.128 − 0.991i)9-s − 0.316i·10-s + 0.101·11-s + (0.375 − 0.330i)12-s − 1.36·13-s + 0.272·14-s + (0.335 − 0.295i)15-s + 0.250·16-s + 1.83·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.989 - 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821502 + 0.0598259i\)
\(L(\frac12)\) \(\approx\) \(0.821502 + 0.0598259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.30 - 1.14i)T \)
5 \( 1 + T \)
23 \( 1 + (-2.61 + 4.02i)T \)
good7 \( 1 + 1.02iT - 7T^{2} \)
11 \( 1 - 0.336T + 11T^{2} \)
13 \( 1 + 4.92T + 13T^{2} \)
17 \( 1 - 7.57T + 17T^{2} \)
19 \( 1 + 3.95iT - 19T^{2} \)
29 \( 1 + 2.20iT - 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 - 0.378iT - 37T^{2} \)
41 \( 1 + 7.82iT - 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 - 5.34iT - 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 - 2.53iT - 59T^{2} \)
61 \( 1 + 12.9iT - 61T^{2} \)
67 \( 1 + 3.67iT - 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + 7.22T + 73T^{2} \)
79 \( 1 + 4.78iT - 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−10.31896371103222278425609426460, −9.723159852326056954765262928718, −8.816964308160793782152260122980, −7.60235226294160792606103055023, −7.05333561260716105197396797354, −5.93483131358960284219202436942, −5.02003340014304622333077330053, −4.33138405856374832498682292013, −3.13066995536164747491758359751, −0.58018778887774996648560405132, 1.16432038828760343343069816884, 2.53207719992099672454670099547, 3.80127611885911323711968138475, 5.17724198006961656668004792706, 5.65807881336198901571632183181, 7.14908403293520868611882689955, 7.68842881483037530961324620141, 8.735340430751492618609522488480, 9.959297692484514745312276862205, 10.40076190631592837985658150199

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