Properties

Label 2-690-23.22-c2-0-17
Degree $2$
Conductor $690$
Sign $0.864 + 0.502i$
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  − 1.41·2-s + 1.73·3-s + 2.00·4-s − 2.23i·5-s − 2.44·6-s + 12.3i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s − 16.3i·11-s + 3.46·12-s − 0.736·13-s − 17.4i·14-s − 3.87i·15-s + 4.00·16-s − 27.6i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.500·4-s − 0.447i·5-s − 0.408·6-s + 1.75i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s − 1.49i·11-s + 0.288·12-s − 0.0566·13-s − 1.24i·14-s − 0.258i·15-s + 0.250·16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.864 + 0.502i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.864 + 0.502i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.585977053\)
\(L(\frac12)\) \(\approx\) \(1.585977053\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - 1.73T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (11.5 - 19.8i)T \)
good7 \( 1 - 12.3iT - 49T^{2} \)
11 \( 1 + 16.3iT - 121T^{2} \)
13 \( 1 + 0.736T + 169T^{2} \)
17 \( 1 + 27.6iT - 289T^{2} \)
19 \( 1 + 16.9iT - 361T^{2} \)
29 \( 1 - 45.7T + 841T^{2} \)
31 \( 1 - 55.8T + 961T^{2} \)
37 \( 1 - 4.22iT - 1.36e3T^{2} \)
41 \( 1 - 23.5T + 1.68e3T^{2} \)
43 \( 1 - 8.31iT - 1.84e3T^{2} \)
47 \( 1 - 19.7T + 2.20e3T^{2} \)
53 \( 1 - 28.0iT - 2.80e3T^{2} \)
59 \( 1 - 61.9T + 3.48e3T^{2} \)
61 \( 1 + 56.2iT - 3.72e3T^{2} \)
67 \( 1 + 63.8iT - 4.48e3T^{2} \)
71 \( 1 - 22.3T + 5.04e3T^{2} \)
73 \( 1 - 93.7T + 5.32e3T^{2} \)
79 \( 1 - 29.5iT - 6.24e3T^{2} \)
83 \( 1 - 119. iT - 6.88e3T^{2} \)
89 \( 1 + 151. iT - 7.92e3T^{2} \)
97 \( 1 + 89.5iT - 9.40e3T^{2} \)
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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

−9.768716727200324425465341128024, −9.231703453533000728041704306370, −8.513005228649156858527417845301, −8.054942334785008425129215173303, −6.70156346040947193456492327897, −5.77215570274617556736963008053, −4.85489230302420269400422786519, −3.05375646416629006726767454261, −2.47607157079192695093547044128, −0.792051012330145022921473584883, 1.14885499589915474953361038769, 2.35939090815727214668928616857, 3.81840289227324038219513477023, 4.46709120141916845970090975681, 6.37475130957463309430344241227, 6.96804280499379622383791989307, 7.86204650982616014150792866226, 8.366087544116413364544530657856, 9.821964048962017347107638057403, 10.26332658596951619291137728476

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