Properties

Label 2-690-115.114-c2-0-18
Degree $2$
Conductor $690$
Sign $0.887 + 0.461i$
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  − 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (4.99 + 0.134i)5-s − 2.44·6-s − 11.0·7-s + 2.82i·8-s − 2.99·9-s + (0.190 − 7.06i)10-s + 9.44i·11-s + 3.46i·12-s + 22.3i·13-s + 15.6i·14-s + (0.233 − 8.65i)15-s + 4.00·16-s + 29.3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (0.999 + 0.0269i)5-s − 0.408·6-s − 1.58·7-s + 0.353i·8-s − 0.333·9-s + (0.0190 − 0.706i)10-s + 0.858i·11-s + 0.288i·12-s + 1.72i·13-s + 1.11i·14-s + (0.0155 − 0.577i)15-s + 0.250·16-s + 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.887 + 0.461i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.887 + 0.461i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.665171368\)
\(L(\frac12)\) \(\approx\) \(1.665171368\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + 1.73iT \)
5 \( 1 + (-4.99 - 0.134i)T \)
23 \( 1 + (-20.6 - 10.0i)T \)
good7 \( 1 + 11.0T + 49T^{2} \)
11 \( 1 - 9.44iT - 121T^{2} \)
13 \( 1 - 22.3iT - 169T^{2} \)
17 \( 1 - 29.3T + 289T^{2} \)
19 \( 1 + 31.8iT - 361T^{2} \)
29 \( 1 - 21.8T + 841T^{2} \)
31 \( 1 + 33.9T + 961T^{2} \)
37 \( 1 - 7.59T + 1.36e3T^{2} \)
41 \( 1 - 55.7T + 1.68e3T^{2} \)
43 \( 1 + 22.0T + 1.84e3T^{2} \)
47 \( 1 - 77.2iT - 2.20e3T^{2} \)
53 \( 1 + 39.1T + 2.80e3T^{2} \)
59 \( 1 - 70.2T + 3.48e3T^{2} \)
61 \( 1 - 31.9iT - 3.72e3T^{2} \)
67 \( 1 - 21.9T + 4.48e3T^{2} \)
71 \( 1 + 31.2T + 5.04e3T^{2} \)
73 \( 1 - 64.0iT - 5.32e3T^{2} \)
79 \( 1 - 8.93iT - 6.24e3T^{2} \)
83 \( 1 - 6.63T + 6.88e3T^{2} \)
89 \( 1 + 153. iT - 7.92e3T^{2} \)
97 \( 1 - 71.0T + 9.40e3T^{2} \)
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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

−9.981874640036896128445482430683, −9.419367885945565371775249197431, −9.011115001727407044883185273348, −7.29278432444615078524794767476, −6.72132967873692842662758617850, −5.78884180274510819057945953671, −4.63228125214806638642787332323, −3.22096169424883781089225150838, −2.36433346721932089537466772419, −1.11615921333200950098986118551, 0.72005684250205394920292711542, 3.04368999033989255146561015606, 3.55512960583277148514522069155, 5.43834558561914446701793062931, 5.68658093344747612000384361061, 6.52413767790166507239918565829, 7.77731542656729496249253431930, 8.641613966840347910869895773254, 9.641664035208406958689214288758, 10.09524398058894532092843086551

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