Properties

Label 2-690-115.114-c2-0-25
Degree $2$
Conductor $690$
Sign $0.698 - 0.716i$
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 + (4.27 − 2.60i)5-s + 2.44·6-s + 3.19·7-s − 2.82i·8-s − 2.99·9-s + (3.67 + 6.03i)10-s + 11.2i·11-s + 3.46i·12-s + 14.7i·13-s + 4.51i·14-s + (−4.50 − 7.39i)15-s + 4.00·16-s + 10.6·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (0.854 − 0.520i)5-s + 0.408·6-s + 0.456·7-s − 0.353i·8-s − 0.333·9-s + (0.367 + 0.603i)10-s + 1.01i·11-s + 0.288i·12-s + 1.13i·13-s + 0.322i·14-s + (−0.300 − 0.493i)15-s + 0.250·16-s + 0.628·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.698 - 0.716i$
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: \((2,\ 690,\ (\ :1),\ 0.698 - 0.716i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.119431002\)
\(L(\frac12)\) \(\approx\) \(2.119431002\)
\(L(2)\) 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 \( 1 + (-4.27 + 2.60i)T \)
23 \( 1 + (22.2 - 5.71i)T \)
good7 \( 1 - 3.19T + 49T^{2} \)
11 \( 1 - 11.2iT - 121T^{2} \)
13 \( 1 - 14.7iT - 169T^{2} \)
17 \( 1 - 10.6T + 289T^{2} \)
19 \( 1 - 9.81iT - 361T^{2} \)
29 \( 1 - 24.7T + 841T^{2} \)
31 \( 1 - 37.7T + 961T^{2} \)
37 \( 1 - 45.5T + 1.36e3T^{2} \)
41 \( 1 - 57.0T + 1.68e3T^{2} \)
43 \( 1 + 15.9T + 1.84e3T^{2} \)
47 \( 1 + 12.7iT - 2.20e3T^{2} \)
53 \( 1 + 62.0T + 2.80e3T^{2} \)
59 \( 1 - 76.6T + 3.48e3T^{2} \)
61 \( 1 + 39.0iT - 3.72e3T^{2} \)
67 \( 1 - 18.2T + 4.48e3T^{2} \)
71 \( 1 - 94.9T + 5.04e3T^{2} \)
73 \( 1 + 8.59iT - 5.32e3T^{2} \)
79 \( 1 - 96.9iT - 6.24e3T^{2} \)
83 \( 1 + 69.0T + 6.88e3T^{2} \)
89 \( 1 - 91.6iT - 7.92e3T^{2} \)
97 \( 1 + 99.7T + 9.40e3T^{2} \)
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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

−9.914289174880963670126416265484, −9.596134358059858973812135970380, −8.415420825112988286107771769151, −7.81790262301587669339929887496, −6.72842530228486217560081457391, −6.07691368909091956998911878712, −5.03585450665923417755544074035, −4.21335372813377141758368704637, −2.29713304807717950388901291382, −1.24289405535891567771376382253, 0.901361855681399265982468238161, 2.54222812014888819958683585753, 3.26899522696119133935276656573, 4.57206362235782595864583265587, 5.58616426277081743779935390228, 6.27381616572871895167412495112, 7.85860225148967827705688465303, 8.538489824181166409423542499034, 9.658838327408738786519003332998, 10.16609969486433793631812641325

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