Properties

Label 2-690-23.22-c2-0-3
Degree $2$
Conductor $690$
Sign $-0.135 - 0.990i$
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 + 5.52i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s + 3.99i·11-s − 3.46·12-s + 8.18·13-s − 7.81i·14-s + 3.87i·15-s + 4.00·16-s − 17.3i·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 + 0.789i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s + 0.363i·11-s − 0.288·12-s + 0.629·13-s − 0.558i·14-s + 0.258i·15-s + 0.250·16-s − 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.135 - 0.990i$
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: \((2,\ 690,\ (\ :1),\ -0.135 - 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6855221208\)
\(L(\frac12)\) \(\approx\) \(0.6855221208\)
\(L(2)\) 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 \( 1 + (22.7 - 3.11i)T \)
good7 \( 1 - 5.52iT - 49T^{2} \)
11 \( 1 - 3.99iT - 121T^{2} \)
13 \( 1 - 8.18T + 169T^{2} \)
17 \( 1 + 17.3iT - 289T^{2} \)
19 \( 1 - 11.0iT - 361T^{2} \)
29 \( 1 - 10.3T + 841T^{2} \)
31 \( 1 + 20.6T + 961T^{2} \)
37 \( 1 - 39.0iT - 1.36e3T^{2} \)
41 \( 1 - 73.4T + 1.68e3T^{2} \)
43 \( 1 + 52.2iT - 1.84e3T^{2} \)
47 \( 1 + 74.9T + 2.20e3T^{2} \)
53 \( 1 - 18.2iT - 2.80e3T^{2} \)
59 \( 1 + 6.77T + 3.48e3T^{2} \)
61 \( 1 - 115. iT - 3.72e3T^{2} \)
67 \( 1 - 110. iT - 4.48e3T^{2} \)
71 \( 1 - 30.5T + 5.04e3T^{2} \)
73 \( 1 + 89.5T + 5.32e3T^{2} \)
79 \( 1 - 141. iT - 6.24e3T^{2} \)
83 \( 1 - 66.8iT - 6.88e3T^{2} \)
89 \( 1 - 54.8iT - 7.92e3T^{2} \)
97 \( 1 + 8.89iT - 9.40e3T^{2} \)
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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

−10.36992681280742918910720442252, −9.622947676972952578313235568534, −8.844168625377800611953506779876, −8.016714085442868814539600599674, −7.04606344516919938539837700555, −6.00802963417444702701638793915, −5.31399202755941675908695458009, −4.05112157487361118759017510492, −2.50202274909120223915067816199, −1.20146110839447914743611823168, 0.37194185236309882170280513909, 1.76080636411348264330578283477, 3.35871193997169000003956097688, 4.39932216330615887603197859323, 5.89638740626332874150850403128, 6.46525807345768441756089222527, 7.47737132130850622412472856044, 8.198405916716207272463887149796, 9.291047334682664686241738070533, 10.16758501041092267587355728081

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