Properties

Label 2-690-5.2-c2-0-31
Degree $2$
Conductor $690$
Sign $0.886 - 0.462i$
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 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (3.78 − 3.26i)5-s − 2.44·6-s + (3.72 + 3.72i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (7.05 + 0.517i)10-s + 11.8·11-s + (−2.44 − 2.44i)12-s + (17.6 − 17.6i)13-s + 7.44i·14-s + (−0.633 + 8.63i)15-s − 4·16-s + (−20.6 − 20.6i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (0.756 − 0.653i)5-s − 0.408·6-s + (0.531 + 0.531i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.705 + 0.0517i)10-s + 1.07·11-s + (−0.204 − 0.204i)12-s + (1.35 − 1.35i)13-s + 0.531i·14-s + (−0.0422 + 0.575i)15-s − 0.250·16-s + (−1.21 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.886 - 0.462i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.754982929\)
\(L(\frac12)\) \(\approx\) \(2.754982929\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (-3.78 + 3.26i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (-3.72 - 3.72i)T + 49iT^{2} \)
11 \( 1 - 11.8T + 121T^{2} \)
13 \( 1 + (-17.6 + 17.6i)T - 169iT^{2} \)
17 \( 1 + (20.6 + 20.6i)T + 289iT^{2} \)
19 \( 1 + 6.19iT - 361T^{2} \)
29 \( 1 - 0.0875iT - 841T^{2} \)
31 \( 1 + 0.753T + 961T^{2} \)
37 \( 1 + (-28.5 - 28.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 57.1T + 1.68e3T^{2} \)
43 \( 1 + (1.66 - 1.66i)T - 1.84e3iT^{2} \)
47 \( 1 + (-40.7 - 40.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-7.42 + 7.42i)T - 2.80e3iT^{2} \)
59 \( 1 - 106. iT - 3.48e3T^{2} \)
61 \( 1 + 91.1T + 3.72e3T^{2} \)
67 \( 1 + (-22.3 - 22.3i)T + 4.48e3iT^{2} \)
71 \( 1 + 123.T + 5.04e3T^{2} \)
73 \( 1 + (-14.6 + 14.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 22.2iT - 6.24e3T^{2} \)
83 \( 1 + (-4.59 + 4.59i)T - 6.88e3iT^{2} \)
89 \( 1 - 30.0iT - 7.92e3T^{2} \)
97 \( 1 + (-120. - 120. i)T + 9.40e3iT^{2} \)
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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

−10.39074477894690792345558566832, −9.051117651399167193567250514876, −8.928466612668525274473827900393, −7.69821721686876212734854523877, −6.36479466369708617365495254970, −5.85177889312843309677041016473, −4.95424320844831748312526961598, −4.14943683710939285257644589268, −2.71049462218562175804249254697, −1.04965597769330366872286233151, 1.36978101599309902751855687564, 2.06699118631096059828100881401, 3.78864546435008091140205256570, 4.43350931287037499303041256039, 6.07941373767531796924980000437, 6.27563778771978131088631270227, 7.26365320101820984103505287343, 8.671992768864751628194084626610, 9.413903121880286887131245559527, 10.65119909500570518236755509155

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