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Properties

Label 2-690-5.3-c2-0-0
Degree $2$
Conductor $690$
Sign $-0.680 - 0.732i$
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 + (−2.46 − 4.34i)5-s + 2.44·6-s + (−1.62 + 1.62i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.81 + 1.88i)10-s + 0.00768·11-s + (−2.44 + 2.44i)12-s + (−7.99 − 7.99i)13-s − 3.24i·14-s + (−2.30 + 8.34i)15-s − 4·16-s + (5.07 − 5.07i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.493 − 0.869i)5-s + 0.408·6-s + (−0.231 + 0.231i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.681 + 0.188i)10-s + 0.000698·11-s + (−0.204 + 0.204i)12-s + (−0.614 − 0.614i)13-s − 0.231i·14-s + (−0.153 + 0.556i)15-s − 0.250·16-s + (0.298 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.680 - 0.732i$
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: \((2,\ 690,\ (\ :1),\ -0.680 - 0.732i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.08544171934\)
\(L(\frac12)\) \(\approx\) \(0.08544171934\)
\(L(2)\) 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 \( 1 + (2.46 + 4.34i)T \)
23 \( 1 + (-3.39 - 3.39i)T \)
good7 \( 1 + (1.62 - 1.62i)T - 49iT^{2} \)
11 \( 1 - 0.00768T + 121T^{2} \)
13 \( 1 + (7.99 + 7.99i)T + 169iT^{2} \)
17 \( 1 + (-5.07 + 5.07i)T - 289iT^{2} \)
19 \( 1 + 29.4iT - 361T^{2} \)
29 \( 1 + 24.1iT - 841T^{2} \)
31 \( 1 + 27.9T + 961T^{2} \)
37 \( 1 + (17.5 - 17.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 9.62T + 1.68e3T^{2} \)
43 \( 1 + (-33.5 - 33.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (48.5 - 48.5i)T - 2.20e3iT^{2} \)
53 \( 1 + (-41.9 - 41.9i)T + 2.80e3iT^{2} \)
59 \( 1 - 85.5iT - 3.48e3T^{2} \)
61 \( 1 + 10.2T + 3.72e3T^{2} \)
67 \( 1 + (80.3 - 80.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 49.4T + 5.04e3T^{2} \)
73 \( 1 + (99.6 + 99.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 114. iT - 6.24e3T^{2} \)
83 \( 1 + (-109. - 109. i)T + 6.88e3iT^{2} \)
89 \( 1 - 16.9iT - 7.92e3T^{2} \)
97 \( 1 + (73.9 - 73.9i)T - 9.40e3iT^{2} \)
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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

−10.55399587333361042323548367234, −9.434781078905758398011227421331, −8.908253272696218380416519045450, −7.78153373390567924927433694851, −7.33655009477499643080471848617, −6.16184627134298452218754432812, −5.27328280276340110868212467576, −4.47116084490160151639776346048, −2.74978806056362398612233444373, −1.08440569722699832788619346282, 0.04423360926378447886081997306, 1.92822902415556300252336075041, 3.38497493247330236267610862744, 4.00503800215817125290971658152, 5.38117276002106870175333823196, 6.59762246391703232348478841646, 7.32139490397352629399277778550, 8.259965229170175627510674251363, 9.322362162150234438659570046422, 10.21204308510444988006693169402

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