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Properties

Label 2-690-5.2-c2-0-9
Degree $2$
Conductor $690$
Sign $0.960 - 0.277i$
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 + (−4.35 + 2.45i)5-s + 2.44·6-s + (−3.40 − 3.40i)7-s + (2 − 2i)8-s − 2.99i·9-s + (6.81 + 1.90i)10-s − 17.0·11-s + (−2.44 − 2.44i)12-s + (9.51 − 9.51i)13-s + 6.81i·14-s + (2.33 − 8.34i)15-s − 4·16-s + (−1.23 − 1.23i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.871 + 0.490i)5-s + 0.408·6-s + (−0.486 − 0.486i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.681 + 0.190i)10-s − 1.55·11-s + (−0.204 − 0.204i)12-s + (0.732 − 0.732i)13-s + 0.486i·14-s + (0.155 − 0.556i)15-s − 0.250·16-s + (−0.0724 − 0.0724i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.960 - 0.277i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.960 - 0.277i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5781073824\)
\(L(\frac12)\) \(\approx\) \(0.5781073824\)
\(L(2)\) 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 \( 1 + (4.35 - 2.45i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (3.40 + 3.40i)T + 49iT^{2} \)
11 \( 1 + 17.0T + 121T^{2} \)
13 \( 1 + (-9.51 + 9.51i)T - 169iT^{2} \)
17 \( 1 + (1.23 + 1.23i)T + 289iT^{2} \)
19 \( 1 + 10.6iT - 361T^{2} \)
29 \( 1 - 16.4iT - 841T^{2} \)
31 \( 1 + 21.1T + 961T^{2} \)
37 \( 1 + (-0.589 - 0.589i)T + 1.36e3iT^{2} \)
41 \( 1 - 44.7T + 1.68e3T^{2} \)
43 \( 1 + (49.7 - 49.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-59.0 - 59.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (38.1 - 38.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 16.5iT - 3.48e3T^{2} \)
61 \( 1 - 85.3T + 3.72e3T^{2} \)
67 \( 1 + (-87.1 - 87.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 18.0T + 5.04e3T^{2} \)
73 \( 1 + (17.4 - 17.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 67.2iT - 6.24e3T^{2} \)
83 \( 1 + (-8.41 + 8.41i)T - 6.88e3iT^{2} \)
89 \( 1 + 44.8iT - 7.92e3T^{2} \)
97 \( 1 + (-84.0 - 84.0i)T + 9.40e3iT^{2} \)
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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

−10.55515148836357807592447288920, −9.706906128565053556038583187798, −8.563671248364239528184569803167, −7.75545401966969734573062501431, −7.02132978586775391694810860290, −5.80418038215087910681104436376, −4.61538160990410555838341643924, −3.53348060436159662535045594669, −2.75319958612297497714413829786, −0.63628976319314413333489020561, 0.46472936332166628577373353297, 2.13121944002756264287817039863, 3.73267384726656974773074423029, 5.03056519549730502141900888183, 5.80199064302433329593745581520, 6.79714025135540256450440701256, 7.72047135649954985026535682554, 8.341334619511213744212723585449, 9.152796568907248582232640317194, 10.23579349420142895066533649252

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