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Properties

Label 2-690-23.22-c2-0-28
Degree $2$
Conductor $690$
Sign $0.415 + 0.909i$
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 − 11.4i·7-s + 2.82·8-s + 2.99·9-s + 3.16i·10-s − 13.8i·11-s + 3.46·12-s − 2.98·13-s − 16.1i·14-s + 3.87i·15-s + 4.00·16-s − 15.3i·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 − 1.63i·7-s + 0.353·8-s + 0.333·9-s + 0.316i·10-s − 1.26i·11-s + 0.288·12-s − 0.229·13-s − 1.15i·14-s + 0.258i·15-s + 0.250·16-s − 0.902i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.415 + 0.909i$
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: \((2,\ 690,\ (\ :1),\ 0.415 + 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.306954428\)
\(L(\frac12)\) \(\approx\) \(3.306954428\)
\(L(2)\) 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 \( 1 + (20.9 - 9.54i)T \)
good7 \( 1 + 11.4iT - 49T^{2} \)
11 \( 1 + 13.8iT - 121T^{2} \)
13 \( 1 + 2.98T + 169T^{2} \)
17 \( 1 + 15.3iT - 289T^{2} \)
19 \( 1 - 2.52iT - 361T^{2} \)
29 \( 1 - 11.6T + 841T^{2} \)
31 \( 1 - 2.71T + 961T^{2} \)
37 \( 1 + 31.7iT - 1.36e3T^{2} \)
41 \( 1 - 36.3T + 1.68e3T^{2} \)
43 \( 1 - 19.9iT - 1.84e3T^{2} \)
47 \( 1 - 52.2T + 2.20e3T^{2} \)
53 \( 1 - 59.4iT - 2.80e3T^{2} \)
59 \( 1 - 81.6T + 3.48e3T^{2} \)
61 \( 1 + 42.2iT - 3.72e3T^{2} \)
67 \( 1 - 58.5iT - 4.48e3T^{2} \)
71 \( 1 - 100.T + 5.04e3T^{2} \)
73 \( 1 + 16.7T + 5.32e3T^{2} \)
79 \( 1 - 60.6iT - 6.24e3T^{2} \)
83 \( 1 - 44.1iT - 6.88e3T^{2} \)
89 \( 1 + 106. iT - 7.92e3T^{2} \)
97 \( 1 - 37.9iT - 9.40e3T^{2} \)
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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

−10.29785610442972069878567367554, −9.371884744597389297851318091759, −8.087301869694095493062593625228, −7.42409871221358735361074231868, −6.65248069783260819397474830922, −5.57335370108869571556068739358, −4.24976444546692819097509731946, −3.61050229867733080940969783708, −2.58035201809167015526775361372, −0.873166318449309637390361989273, 1.88801363970445219204389310241, 2.58459044691275424028520434787, 3.98593108821641336146643578154, 4.91329873485647169436330403890, 5.80125571419841412356635207230, 6.76020072262670137835911807859, 7.937619516093022882347302105224, 8.637807266427061997153645380035, 9.524972217004685930049625468228, 10.29632706551425032334079882125

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