Properties

Label 2-690-5.3-c2-0-19
Degree $2$
Conductor $690$
Sign $0.508 + 0.860i$
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 + (1.48 + 4.77i)5-s + 2.44·6-s + (−9.54 + 9.54i)7-s + (2 + 2i)8-s + 2.99i·9-s + (−6.26 − 3.28i)10-s − 15.2·11-s + (−2.44 + 2.44i)12-s + (−8.82 − 8.82i)13-s − 19.0i·14-s + (4.02 − 7.66i)15-s − 4·16-s + (5.29 − 5.29i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (0.297 + 0.954i)5-s + 0.408·6-s + (−1.36 + 1.36i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.626 − 0.328i)10-s − 1.38·11-s + (−0.204 + 0.204i)12-s + (−0.678 − 0.678i)13-s − 1.36i·14-s + (0.268 − 0.511i)15-s − 0.250·16-s + (0.311 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.508 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.508 + 0.860i$
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: \((2,\ 690,\ (\ :1),\ 0.508 + 0.860i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2589028396\)
\(L(\frac12)\) \(\approx\) \(0.2589028396\)
\(L(2)\) 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 \( 1 + (-1.48 - 4.77i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (9.54 - 9.54i)T - 49iT^{2} \)
11 \( 1 + 15.2T + 121T^{2} \)
13 \( 1 + (8.82 + 8.82i)T + 169iT^{2} \)
17 \( 1 + (-5.29 + 5.29i)T - 289iT^{2} \)
19 \( 1 - 0.807iT - 361T^{2} \)
29 \( 1 + 14.0iT - 841T^{2} \)
31 \( 1 - 29.6T + 961T^{2} \)
37 \( 1 + (6.74 - 6.74i)T - 1.36e3iT^{2} \)
41 \( 1 - 70.9T + 1.68e3T^{2} \)
43 \( 1 + (-36.3 - 36.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (32.5 - 32.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (41.7 + 41.7i)T + 2.80e3iT^{2} \)
59 \( 1 + 103. iT - 3.48e3T^{2} \)
61 \( 1 - 63.7T + 3.72e3T^{2} \)
67 \( 1 + (31.8 - 31.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 61.7T + 5.04e3T^{2} \)
73 \( 1 + (-80.7 - 80.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 132. iT - 6.24e3T^{2} \)
83 \( 1 + (59.6 + 59.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 87.8iT - 7.92e3T^{2} \)
97 \( 1 + (-71.4 + 71.4i)T - 9.40e3iT^{2} \)
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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

−9.896527350659777396104948332116, −9.535310548724715501087251630792, −8.184266384551577118878021408810, −7.47586998472261079047778124329, −6.47963631270921880638421663373, −5.88509603801742699342045341455, −5.15675454555104635229721284492, −2.96842235425101773891035979722, −2.43955324293379676695734630844, −0.14763301494521697769852231252, 0.889760812329840278619189493661, 2.65069273198130671056461268452, 3.91464008459052918497381587499, 4.73616267855345823366043762072, 5.90135654561546059169969954723, 7.02342263946279135182954276941, 7.83597716426055059701565657655, 9.017615747446551776339038225212, 9.748578373834742293303170674478, 10.28061030551962504553157302004

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