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Properties

Label 2-690-3.2-c2-0-41
Degree $2$
Conductor $690$
Sign $0.374 - 0.927i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (2.78 + 1.12i)3-s − 2.00·4-s + 2.23i·5-s + (−1.58 + 3.93i)6-s + 9.71·7-s − 2.82i·8-s + (6.47 + 6.24i)9-s − 3.16·10-s − 16.7i·11-s + (−5.56 − 2.24i)12-s + 14.2·13-s + 13.7i·14-s + (−2.51 + 6.22i)15-s + 4.00·16-s − 8.32i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.927 + 0.374i)3-s − 0.500·4-s + 0.447i·5-s + (−0.264 + 0.655i)6-s + 1.38·7-s − 0.353i·8-s + (0.719 + 0.694i)9-s − 0.316·10-s − 1.52i·11-s + (−0.463 − 0.187i)12-s + 1.09·13-s + 0.981i·14-s + (−0.167 + 0.414i)15-s + 0.250·16-s − 0.489i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.374 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.053098753\)
\(L(\frac12)\) \(\approx\) \(3.053098753\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-2.78 - 1.12i)T \)
5 \( 1 - 2.23iT \)
23 \( 1 + 4.79iT \)
good7 \( 1 - 9.71T + 49T^{2} \)
11 \( 1 + 16.7iT - 121T^{2} \)
13 \( 1 - 14.2T + 169T^{2} \)
17 \( 1 + 8.32iT - 289T^{2} \)
19 \( 1 - 18.7T + 361T^{2} \)
29 \( 1 + 8.35iT - 841T^{2} \)
31 \( 1 - 12.7T + 961T^{2} \)
37 \( 1 + 48.4T + 1.36e3T^{2} \)
41 \( 1 + 33.7iT - 1.68e3T^{2} \)
43 \( 1 + 41.8T + 1.84e3T^{2} \)
47 \( 1 - 25.4iT - 2.20e3T^{2} \)
53 \( 1 - 64.6iT - 2.80e3T^{2} \)
59 \( 1 - 35.0iT - 3.48e3T^{2} \)
61 \( 1 + 57.1T + 3.72e3T^{2} \)
67 \( 1 + 61.2T + 4.48e3T^{2} \)
71 \( 1 - 99.0iT - 5.04e3T^{2} \)
73 \( 1 - 1.71T + 5.32e3T^{2} \)
79 \( 1 - 118.T + 6.24e3T^{2} \)
83 \( 1 - 13.2iT - 6.88e3T^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 + 166.T + 9.40e3T^{2} \)
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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

−10.46098658369046569297918698206, −9.222089280544815352494985493717, −8.517976447040079039901309570700, −8.009867745727829280380177381587, −7.13626125236187038582274567043, −5.90120598993137630796904374458, −5.01007395045143692288966543972, −3.89181287200990999001648461585, −2.96158170745022779140111391982, −1.31134411473176858312933970189, 1.39167849193060515053623336891, 1.87974794070540047995276181108, 3.39018056241652952387429076044, 4.40907949857499896222972533972, 5.21076662422807537798859089695, 6.78848484316036365022496272780, 7.86253190790079151386361081511, 8.346551129724765209367077274434, 9.238484243917338781698641025742, 10.01145046990758036221917862066

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