Properties

Label 2-690-23.22-c2-0-16
Degree $2$
Conductor $690$
Sign $0.862 - 0.506i$
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 − 3.80i·7-s + 2.82·8-s + 2.99·9-s + 3.16i·10-s + 16.1i·11-s + 3.46·12-s + 12.9·13-s − 5.38i·14-s + 3.87i·15-s + 4.00·16-s + 10.8i·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 − 0.543i·7-s + 0.353·8-s + 0.333·9-s + 0.316i·10-s + 1.47i·11-s + 0.288·12-s + 0.997·13-s − 0.384i·14-s + 0.258i·15-s + 0.250·16-s + 0.636i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.862 - 0.506i$
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: \((2,\ 690,\ (\ :1),\ 0.862 - 0.506i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.700423969\)
\(L(\frac12)\) \(\approx\) \(3.700423969\)
\(L(2)\) 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 \( 1 + (-11.6 - 19.8i)T \)
good7 \( 1 + 3.80iT - 49T^{2} \)
11 \( 1 - 16.1iT - 121T^{2} \)
13 \( 1 - 12.9T + 169T^{2} \)
17 \( 1 - 10.8iT - 289T^{2} \)
19 \( 1 + 17.1iT - 361T^{2} \)
29 \( 1 + 10.1T + 841T^{2} \)
31 \( 1 - 29.2T + 961T^{2} \)
37 \( 1 + 3.06iT - 1.36e3T^{2} \)
41 \( 1 - 55.3T + 1.68e3T^{2} \)
43 \( 1 + 34.1iT - 1.84e3T^{2} \)
47 \( 1 + 49.2T + 2.20e3T^{2} \)
53 \( 1 + 28.6iT - 2.80e3T^{2} \)
59 \( 1 - 24.2T + 3.48e3T^{2} \)
61 \( 1 - 52.9iT - 3.72e3T^{2} \)
67 \( 1 - 7.48iT - 4.48e3T^{2} \)
71 \( 1 + 121.T + 5.04e3T^{2} \)
73 \( 1 + 132.T + 5.32e3T^{2} \)
79 \( 1 + 18.6iT - 6.24e3T^{2} \)
83 \( 1 - 46.3iT - 6.88e3T^{2} \)
89 \( 1 - 17.2iT - 7.92e3T^{2} \)
97 \( 1 + 31.4iT - 9.40e3T^{2} \)
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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

−10.40292615401875192795668425092, −9.603260030884524922403833464837, −8.551937318092949572154231441114, −7.41582714219712949022664045801, −6.96818183440341653458541507995, −5.86536954268201829642025550307, −4.57921765607657452726580530002, −3.85273562788989951059688149206, −2.75102996707702952499238634251, −1.53293046763256002240672693228, 1.12550915089572913122235056797, 2.66519565146318101237230263021, 3.53030115565785976927725634183, 4.59989633500824537227701222648, 5.75806903746031592102357096224, 6.32957757450938121727727600204, 7.70751014927082902347353964112, 8.517062479991851978285551416134, 9.083820926346191426148600965307, 10.28954451634396862396198644859

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