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Properties

Label 2-690-5.3-c2-0-20
Degree $2$
Conductor $690$
Sign $0.977 + 0.211i$
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 + (−4.51 − 2.15i)5-s − 2.44·6-s + (−8.14 + 8.14i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.66 − 2.36i)10-s − 2.98·11-s + (2.44 − 2.44i)12-s + (−7.61 − 7.61i)13-s − 16.2i·14-s + (−2.89 − 8.16i)15-s − 4·16-s + (15.0 − 15.0i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.902 − 0.430i)5-s − 0.408·6-s + (−1.16 + 1.16i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.666 − 0.236i)10-s − 0.271·11-s + (0.204 − 0.204i)12-s + (−0.585 − 0.585i)13-s − 1.16i·14-s + (−0.192 − 0.544i)15-s − 0.250·16-s + (0.887 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7758709845\)
\(L(\frac12)\) \(\approx\) \(0.7758709845\)
\(L(2)\) 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 \( 1 + (4.51 + 2.15i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (8.14 - 8.14i)T - 49iT^{2} \)
11 \( 1 + 2.98T + 121T^{2} \)
13 \( 1 + (7.61 + 7.61i)T + 169iT^{2} \)
17 \( 1 + (-15.0 + 15.0i)T - 289iT^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
29 \( 1 - 49.2iT - 841T^{2} \)
31 \( 1 + 5.90T + 961T^{2} \)
37 \( 1 + (-41.6 + 41.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 20.3T + 1.68e3T^{2} \)
43 \( 1 + (22.1 + 22.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-60.9 + 60.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (-63.5 - 63.5i)T + 2.80e3iT^{2} \)
59 \( 1 + 15.2iT - 3.48e3T^{2} \)
61 \( 1 + 12.0T + 3.72e3T^{2} \)
67 \( 1 + (-79.9 + 79.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 20.4T + 5.04e3T^{2} \)
73 \( 1 + (0.611 + 0.611i)T + 5.32e3iT^{2} \)
79 \( 1 + 127. iT - 6.24e3T^{2} \)
83 \( 1 + (24.8 + 24.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 87.9iT - 7.92e3T^{2} \)
97 \( 1 + (-15.5 + 15.5i)T - 9.40e3iT^{2} \)
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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

−9.945531702914879380387691069665, −9.140551389725268874886684395892, −8.744262069450254452180890206530, −7.64345348658929785922012416021, −6.99333279670941667357856180188, −5.59349305084027366744735585865, −5.00982763036258154496622187497, −3.49816891882848454534733857819, −2.61411565055807990880897539233, −0.42047406262943188152126370862, 0.896038105229902484064485392277, 2.59104854489696092854109047406, 3.60498951266481451999972950026, 4.22368258262627898628170490853, 6.20067575460068667674523507914, 7.05308482603503519515128717391, 7.73898256693908401179357925680, 8.394192764701213213965468244298, 9.867116096189693943035586029019, 9.982091694728051563449344937159

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