Properties

Label 2-690-15.8-c1-0-20
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: $\chi_{690} (323, \cdot )$
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

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

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