Properties

Label 2-690-115.114-c2-0-21
Degree $2$
Conductor $690$
Sign $0.956 - 0.291i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.956 - 0.291i$
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: \((2,\ 690,\ (\ :1),\ 0.956 - 0.291i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.165299245\)
\(L(\frac12)\) \(\approx\) \(1.165299245\)
\(L(2)\) 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

−10.35399388897851729387308823867, −9.032677841007068246451127477247, −8.552180870087255080197662807672, −7.50408080071896312351835109641, −6.75342068646231465988757267507, −6.25294892522404206289154187761, −4.87910625515803010546345581000, −3.78210410579789552309323643128, −2.68772138400412579746193979405, −0.66324768365280846646287934633, 0.78552363884524943816763207918, 2.60893840696363354431265606904, 3.65752657500266891411744868069, 4.55119116611598164708558036053, 5.30659583385135875708406799812, 6.73701919643317277379850451337, 7.88267745300720450618167354490, 8.635320439373962333743645863250, 9.514769505268247369286714777620, 10.28271883665464609171303739867

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