Properties

Label 2-690-23.22-c2-0-30
Degree $2$
Conductor $690$
Sign $0.0455 + 0.998i$
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 − 7.54i·7-s + 2.82·8-s + 2.99·9-s − 3.16i·10-s − 8.65i·11-s + 3.46·12-s − 19.9·13-s − 10.6i·14-s − 3.87i·15-s + 4.00·16-s + 9.27i·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 − 1.07i·7-s + 0.353·8-s + 0.333·9-s − 0.316i·10-s − 0.786i·11-s + 0.288·12-s − 1.53·13-s − 0.762i·14-s − 0.258i·15-s + 0.250·16-s + 0.545i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0455 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0455 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0455 + 0.998i$
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: \((2,\ 690,\ (\ :1),\ 0.0455 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.072447900\)
\(L(\frac12)\) \(\approx\) \(3.072447900\)
\(L(2)\) 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 \( 1 + (-22.9 + 1.04i)T \)
good7 \( 1 + 7.54iT - 49T^{2} \)
11 \( 1 + 8.65iT - 121T^{2} \)
13 \( 1 + 19.9T + 169T^{2} \)
17 \( 1 - 9.27iT - 289T^{2} \)
19 \( 1 + 35.9iT - 361T^{2} \)
29 \( 1 + 27.4T + 841T^{2} \)
31 \( 1 - 22.3T + 961T^{2} \)
37 \( 1 - 42.9iT - 1.36e3T^{2} \)
41 \( 1 + 39.2T + 1.68e3T^{2} \)
43 \( 1 - 20.8iT - 1.84e3T^{2} \)
47 \( 1 - 66.5T + 2.20e3T^{2} \)
53 \( 1 + 64.5iT - 2.80e3T^{2} \)
59 \( 1 - 90.0T + 3.48e3T^{2} \)
61 \( 1 - 10.7iT - 3.72e3T^{2} \)
67 \( 1 + 58.7iT - 4.48e3T^{2} \)
71 \( 1 + 29.3T + 5.04e3T^{2} \)
73 \( 1 - 95.8T + 5.32e3T^{2} \)
79 \( 1 - 72.0iT - 6.24e3T^{2} \)
83 \( 1 + 106. iT - 6.88e3T^{2} \)
89 \( 1 - 80.0iT - 7.92e3T^{2} \)
97 \( 1 - 103. iT - 9.40e3T^{2} \)
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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

−10.10005589506637474459018084559, −9.205770286897613699425096075037, −8.240329091104198988187624450438, −7.27474715524885187872146114388, −6.68895323978093440554724509055, −5.19821353120655534539408811986, −4.54525807097862440827070132385, −3.46712680588601702866973959883, −2.42476128063933804952206811551, −0.78092350421102357024878559960, 2.02497581148954942660834101714, 2.71438416187035116134376063123, 3.89181657548174995614771074888, 5.04660609831630616897025780375, 5.82436087933519497086320507767, 7.10406549588423994040021077675, 7.59000437257534099177859185918, 8.793698480411661129960427264763, 9.706502466210426896622776831085, 10.34103025987451187245501221888

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