Properties

Label 2-462-21.17-c1-0-26
Degree $2$
Conductor $462$
Sign $-0.817 + 0.575i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.106 − 1.72i)3-s + (0.499 − 0.866i)4-s + (−0.0938 − 0.162i)5-s + (−0.772 − 1.55i)6-s + (−2.64 + 0.0638i)7-s − 0.999i·8-s + (−2.97 − 0.368i)9-s + (−0.162 − 0.0938i)10-s + (−0.866 − 0.5i)11-s + (−1.44 − 0.956i)12-s − 4.70i·13-s + (−2.25 + 1.37i)14-s + (−0.291 + 0.144i)15-s + (−0.5 − 0.866i)16-s + (1.48 − 2.57i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0614 − 0.998i)3-s + (0.249 − 0.433i)4-s + (−0.0419 − 0.0727i)5-s + (−0.315 − 0.632i)6-s + (−0.999 + 0.0241i)7-s − 0.353i·8-s + (−0.992 − 0.122i)9-s + (−0.0514 − 0.0296i)10-s + (−0.261 − 0.150i)11-s + (−0.416 − 0.276i)12-s − 1.30i·13-s + (−0.603 + 0.368i)14-s + (−0.0751 + 0.0374i)15-s + (−0.125 − 0.216i)16-s + (0.360 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.471195 - 1.48899i\)
\(L(\frac12)\) \(\approx\) \(0.471195 - 1.48899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.106 + 1.72i)T \)
7 \( 1 + (2.64 - 0.0638i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (0.0938 + 0.162i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4.70iT - 13T^{2} \)
17 \( 1 + (-1.48 + 2.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.61 + 0.929i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.05 - 1.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.25iT - 29T^{2} \)
31 \( 1 + (-8.78 - 5.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.73 + 4.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.187T + 41T^{2} \)
43 \( 1 - 7.86T + 43T^{2} \)
47 \( 1 + (-2.79 - 4.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.50 + 5.48i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.32 + 4.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.8 + 6.28i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.28 - 2.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (13.8 + 8.02i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.99 + 6.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + (-4.29 - 7.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.00iT - 97T^{2} \)
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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

−10.79042684158216971975515271700, −9.985554427331534205581994488223, −8.842578630860597037319393394313, −7.79390617547022847669090089999, −6.86068930458658903003425590768, −5.97503856824842353550891686706, −5.08745084284783024548124791113, −3.34099137182653176904136044246, −2.65894248077204177645510324764, −0.78623276514770520153541056580, 2.62544259351409991953451153178, 3.76796281501646155146690045897, 4.52417809505774336678736321012, 5.78128892371714941604832515116, 6.50486912879441931409946823349, 7.71393160000541086810561036342, 8.825800474079242172388670511040, 9.689650894289366143012615176262, 10.39320011373531859813610858184, 11.53173650564445879099511883222

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