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Properties

Label 2-690-5.3-c2-0-6
Degree $2$
Conductor $690$
Sign $-0.999 - 0.0142i$
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 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−4.84 − 1.21i)5-s − 2.44·6-s + (−0.117 + 0.117i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.06 − 3.63i)10-s + 6.14·11-s + (2.44 − 2.44i)12-s + (1.79 + 1.79i)13-s − 0.234i·14-s + (−4.44 − 7.43i)15-s − 4·16-s + (5.21 − 5.21i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.969 − 0.243i)5-s − 0.408·6-s + (−0.0167 + 0.0167i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.606 − 0.363i)10-s + 0.558·11-s + (0.204 − 0.204i)12-s + (0.138 + 0.138i)13-s − 0.0167i·14-s + (−0.296 − 0.495i)15-s − 0.250·16-s + (0.306 − 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0142i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.999 - 0.0142i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.999 - 0.0142i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5628122912\)
\(L(\frac12)\) \(\approx\) \(0.5628122912\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (4.84 + 1.21i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (0.117 - 0.117i)T - 49iT^{2} \)
11 \( 1 - 6.14T + 121T^{2} \)
13 \( 1 + (-1.79 - 1.79i)T + 169iT^{2} \)
17 \( 1 + (-5.21 + 5.21i)T - 289iT^{2} \)
19 \( 1 - 28.4iT - 361T^{2} \)
29 \( 1 + 34.1iT - 841T^{2} \)
31 \( 1 + 24.7T + 961T^{2} \)
37 \( 1 + (48.0 - 48.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 62.5T + 1.68e3T^{2} \)
43 \( 1 + (26.2 + 26.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (46.4 - 46.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 2.80e3iT^{2} \)
59 \( 1 - 11.1iT - 3.48e3T^{2} \)
61 \( 1 - 80.2T + 3.72e3T^{2} \)
67 \( 1 + (74.7 - 74.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 5.49T + 5.04e3T^{2} \)
73 \( 1 + (-14.2 - 14.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 66.3iT - 6.24e3T^{2} \)
83 \( 1 + (113. + 113. i)T + 6.88e3iT^{2} \)
89 \( 1 - 35.8iT - 7.92e3T^{2} \)
97 \( 1 + (93.7 - 93.7i)T - 9.40e3iT^{2} \)
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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

−10.41777624374099085833145294111, −9.770225737903323559451342659168, −8.766458840946490363907737879494, −8.202535025121389770074325879557, −7.42626761078142175891195596134, −6.42319894094533318454376203557, −5.26049122627967546882920676067, −4.19914887413346111441350701809, −3.32085639375461327359082239502, −1.52100910233521913215110818883, 0.23209298385481632240493864631, 1.70636235672025109417148914872, 3.12545883094297119202991908322, 3.80714535727268371038723259153, 5.12557007221174032927773944707, 6.81594556618834350467373334427, 7.17005191086870326697770399747, 8.355113870217322020399473872970, 8.771601368131112548148797832726, 9.805203700869768229307237400698

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