Properties

Label 2-690-23.22-c2-0-29
Degree $2$
Conductor $690$
Sign $-0.873 - 0.486i$
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 − 12.6i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s − 6.64i·11-s − 3.46·12-s − 21.6·13-s + 17.8i·14-s + 3.87i·15-s + 4.00·16-s − 26.9i·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 − 1.80i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s − 0.604i·11-s − 0.288·12-s − 1.66·13-s + 1.27i·14-s + 0.258i·15-s + 0.250·16-s − 1.58i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.873 - 0.486i$
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: \((2,\ 690,\ (\ :1),\ -0.873 - 0.486i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4013312863\)
\(L(\frac12)\) \(\approx\) \(0.4013312863\)
\(L(2)\) 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 \( 1 + (11.1 - 20.0i)T \)
good7 \( 1 + 12.6iT - 49T^{2} \)
11 \( 1 + 6.64iT - 121T^{2} \)
13 \( 1 + 21.6T + 169T^{2} \)
17 \( 1 + 26.9iT - 289T^{2} \)
19 \( 1 + 5.80iT - 361T^{2} \)
29 \( 1 - 50.7T + 841T^{2} \)
31 \( 1 - 0.790T + 961T^{2} \)
37 \( 1 + 11.7iT - 1.36e3T^{2} \)
41 \( 1 + 0.0875T + 1.68e3T^{2} \)
43 \( 1 - 78.4iT - 1.84e3T^{2} \)
47 \( 1 - 65.2T + 2.20e3T^{2} \)
53 \( 1 + 35.8iT - 2.80e3T^{2} \)
59 \( 1 + 28.0T + 3.48e3T^{2} \)
61 \( 1 - 18.6iT - 3.72e3T^{2} \)
67 \( 1 + 23.9iT - 4.48e3T^{2} \)
71 \( 1 + 101.T + 5.04e3T^{2} \)
73 \( 1 + 111.T + 5.32e3T^{2} \)
79 \( 1 - 84.0iT - 6.24e3T^{2} \)
83 \( 1 + 137. iT - 6.88e3T^{2} \)
89 \( 1 - 46.7iT - 7.92e3T^{2} \)
97 \( 1 - 20.6iT - 9.40e3T^{2} \)
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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

−9.938037838026898664575157577182, −9.095374817619606491215411139770, −7.70432876387231353604285471193, −7.37520595600197072929836346935, −6.44914977822748347906079490794, −5.09584378200888147668783379832, −4.35092461521566134446309595249, −2.87066498264019962041884980819, −1.08097655645885586346965953995, −0.22203620755118839311185705363, 1.95253732986598042603499786904, 2.72552650869942562917882831124, 4.50491371407584596343169933524, 5.62664086498052947019209597194, 6.32002413827573327372822398734, 7.27455344371303754850173744768, 8.305340688016298680721053681031, 9.023449621481685389637659114097, 10.13387617624058939425636041953, 10.40205163647135305086706397737

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