Properties

Label 2-690-115.22-c1-0-12
Degree $2$
Conductor $690$
Sign $0.192 + 0.981i$
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 + (0.707 − 0.707i)3-s + 1.00i·4-s + (−2.13 + 0.677i)5-s − 1.00·6-s + (−1.05 + 1.05i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (1.98 + 1.02i)10-s + 2.05i·11-s + (0.707 + 0.707i)12-s + (2.75 − 2.75i)13-s + 1.49·14-s + (−1.02 + 1.98i)15-s − 1.00·16-s + (4.76 − 4.76i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (−0.952 + 0.303i)5-s − 0.408·6-s + (−0.400 + 0.400i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.628 + 0.324i)10-s + 0.619i·11-s + (0.204 + 0.204i)12-s + (0.763 − 0.763i)13-s + 0.400·14-s + (−0.265 + 0.512i)15-s − 0.250·16-s + (1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.192 + 0.981i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.815938 - 0.671626i\)
\(L(\frac12)\) \(\approx\) \(0.815938 - 0.671626i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2.13 - 0.677i)T \)
23 \( 1 + (3.76 + 2.97i)T \)
good7 \( 1 + (1.05 - 1.05i)T - 7iT^{2} \)
11 \( 1 - 2.05iT - 11T^{2} \)
13 \( 1 + (-2.75 + 2.75i)T - 13iT^{2} \)
17 \( 1 + (-4.76 + 4.76i)T - 17iT^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
29 \( 1 + 7.96iT - 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + (-2.66 + 2.66i)T - 37iT^{2} \)
41 \( 1 + 8.60T + 41T^{2} \)
43 \( 1 + (-1.45 - 1.45i)T + 43iT^{2} \)
47 \( 1 + (-1.89 - 1.89i)T + 47iT^{2} \)
53 \( 1 + (-3.57 - 3.57i)T + 53iT^{2} \)
59 \( 1 + 3.82iT - 59T^{2} \)
61 \( 1 - 2.76iT - 61T^{2} \)
67 \( 1 + (-7.21 + 7.21i)T - 67iT^{2} \)
71 \( 1 - 0.620T + 71T^{2} \)
73 \( 1 + (-4.77 + 4.77i)T - 73iT^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (-5.71 - 5.71i)T + 83iT^{2} \)
89 \( 1 - 0.705T + 89T^{2} \)
97 \( 1 + (-0.564 + 0.564i)T - 97iT^{2} \)
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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

−10.11250281002017739891096202879, −9.569601225502505285086414475598, −8.337715897772682369918422585484, −7.921729581775153058810555575702, −7.04888097357587194791515183352, −5.97612853826917450046013308137, −4.46544030006832517725100898126, −3.31921931623100542539352851928, −2.57510136468641951879080317354, −0.76083082196446612528939906524, 1.23410083876535946069883570510, 3.34819459031166380509567685548, 4.01277826889587562558162319835, 5.26911528918262252423304642048, 6.37012877332580573754240207373, 7.32292768117554978038749801911, 8.347433965286441976982198990692, 8.572168133111702375921624810855, 9.787037275379602770722821885048, 10.41234432158284393177427196408

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