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Properties

Label 2-690-15.2-c1-0-29
Degree $2$
Conductor $690$
Sign $0.872 + 0.487i$
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  + (−0.707 + 0.707i)2-s + (1.47 − 0.911i)3-s − 1.00i·4-s + (2.23 + 0.104i)5-s + (−0.396 + 1.68i)6-s + (−1.29 − 1.29i)7-s + (0.707 + 0.707i)8-s + (1.33 − 2.68i)9-s + (−1.65 + 1.50i)10-s + 0.254i·11-s + (−0.911 − 1.47i)12-s + (0.686 − 0.686i)13-s + 1.83·14-s + (3.38 − 1.88i)15-s − 1.00·16-s + (2.14 − 2.14i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.850 − 0.526i)3-s − 0.500i·4-s + (0.998 + 0.0466i)5-s + (−0.161 + 0.688i)6-s + (−0.490 − 0.490i)7-s + (0.250 + 0.250i)8-s + (0.445 − 0.895i)9-s + (−0.522 + 0.476i)10-s + 0.0768i·11-s + (−0.263 − 0.425i)12-s + (0.190 − 0.190i)13-s + 0.490·14-s + (0.873 − 0.486i)15-s − 0.250·16-s + (0.520 − 0.520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.872 + 0.487i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.872 + 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72187 - 0.448457i\)
\(L(\frac12)\) \(\approx\) \(1.72187 - 0.448457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.47 + 0.911i)T \)
5 \( 1 + (-2.23 - 0.104i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1.29 + 1.29i)T + 7iT^{2} \)
11 \( 1 - 0.254iT - 11T^{2} \)
13 \( 1 + (-0.686 + 0.686i)T - 13iT^{2} \)
17 \( 1 + (-2.14 + 2.14i)T - 17iT^{2} \)
19 \( 1 - 0.235iT - 19T^{2} \)
29 \( 1 - 0.789T + 29T^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 + (4.51 + 4.51i)T + 37iT^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 + (-2.43 + 2.43i)T - 43iT^{2} \)
47 \( 1 + (2.45 - 2.45i)T - 47iT^{2} \)
53 \( 1 + (-2.31 - 2.31i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + (-1.01 - 1.01i)T + 67iT^{2} \)
71 \( 1 - 1.05iT - 71T^{2} \)
73 \( 1 + (-6.97 + 6.97i)T - 73iT^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 + (-8.88 - 8.88i)T + 83iT^{2} \)
89 \( 1 + 9.41T + 89T^{2} \)
97 \( 1 + (-2.94 - 2.94i)T + 97iT^{2} \)
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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

−9.995667931403017067238349892854, −9.503221936199095872154683219503, −8.713775434877478984608389182832, −7.74948523430409730534347035781, −6.95269856430975436910435145968, −6.25436717056338990132404874708, −5.18044442962015105906386592986, −3.62205056152156345378734399160, −2.42935635181398154453566272752, −1.12332711220925487649350421370, 1.69917734067592483557290027755, 2.72242661836315799364181898561, 3.66149607744528390489482268429, 4.98164147797852364699130902669, 6.06776398891720011192246173276, 7.19684662137270335315754247900, 8.379204407670126632608493855501, 8.952913720087501218712833201735, 9.687240890199686262711976251752, 10.26715678062494633026301367108

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