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Properties

Label 2-690-23.22-c2-0-2
Degree $2$
Conductor $690$
Sign $-0.859 - 0.511i$
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 + 7.96i·7-s + 2.82·8-s + 2.99·9-s + 3.16i·10-s + 13.6i·11-s − 3.46·12-s − 14.0·13-s + 11.2i·14-s − 3.87i·15-s + 4.00·16-s − 17.2i·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 + 1.13i·7-s + 0.353·8-s + 0.333·9-s + 0.316i·10-s + 1.24i·11-s − 0.288·12-s − 1.07·13-s + 0.804i·14-s − 0.258i·15-s + 0.250·16-s − 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.859 - 0.511i$
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: \((2,\ 690,\ (\ :1),\ -0.859 - 0.511i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.250251007\)
\(L(\frac12)\) \(\approx\) \(1.250251007\)
\(L(2)\) 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 \( 1 + (11.7 - 19.7i)T \)
good7 \( 1 - 7.96iT - 49T^{2} \)
11 \( 1 - 13.6iT - 121T^{2} \)
13 \( 1 + 14.0T + 169T^{2} \)
17 \( 1 + 17.2iT - 289T^{2} \)
19 \( 1 + 26.4iT - 361T^{2} \)
29 \( 1 + 54.4T + 841T^{2} \)
31 \( 1 + 22.1T + 961T^{2} \)
37 \( 1 - 32.2iT - 1.36e3T^{2} \)
41 \( 1 - 9.59T + 1.68e3T^{2} \)
43 \( 1 - 65.3iT - 1.84e3T^{2} \)
47 \( 1 - 82.5T + 2.20e3T^{2} \)
53 \( 1 - 21.0iT - 2.80e3T^{2} \)
59 \( 1 + 105.T + 3.48e3T^{2} \)
61 \( 1 - 33.4iT - 3.72e3T^{2} \)
67 \( 1 + 34.9iT - 4.48e3T^{2} \)
71 \( 1 - 65.1T + 5.04e3T^{2} \)
73 \( 1 + 126.T + 5.32e3T^{2} \)
79 \( 1 - 45.0iT - 6.24e3T^{2} \)
83 \( 1 - 10.3iT - 6.88e3T^{2} \)
89 \( 1 + 20.8iT - 7.92e3T^{2} \)
97 \( 1 - 146. iT - 9.40e3T^{2} \)
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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

−10.89684274112555143309135654437, −9.664647998931643039572505949699, −9.268309686308565128329182645025, −7.48897305382404616934127780738, −7.16148461058475848795999455478, −5.96800014249526968432482719877, −5.18211502450168102490868305859, −4.43849761513870668337180910348, −2.88821527023860123806342592967, −2.02590749520182125055933703151, 0.35037453752131581765729595129, 1.85068159249664124207553502102, 3.65909513589905395515225323674, 4.20748059761502212427456440890, 5.50254766229803500485051494831, 6.01347744924499095558567460570, 7.23781616695409529597398170569, 7.88679927183695120576687795036, 9.084239681332635906150398364433, 10.38224717707468963407181408940

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