Properties

Label 2-462-11.3-c1-0-7
Degree $2$
Conductor $462$
Sign $0.528 + 0.849i$
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  + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (1.69 + 1.23i)5-s + (−0.809 − 0.587i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 2.10·10-s + (3.19 − 0.871i)11-s − 12-s + (4.47 − 3.25i)13-s + (0.309 + 0.951i)14-s + (0.649 − 1.99i)15-s + (−0.809 − 0.587i)16-s + (−1.5 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.760 + 0.552i)5-s + (−0.330 − 0.239i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + 0.664·10-s + (0.964 − 0.262i)11-s − 0.288·12-s + (1.24 − 0.902i)13-s + (0.0825 + 0.254i)14-s + (0.167 − 0.515i)15-s + (−0.202 − 0.146i)16-s + (−0.363 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.528 + 0.849i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.528 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83928 - 1.02208i\)
\(L(\frac12)\) \(\approx\) \(1.83928 - 1.02208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-3.19 + 0.871i)T \)
good5 \( 1 + (-1.69 - 1.23i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.47 + 3.25i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.5 + 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.283 - 0.871i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.37T + 23T^{2} \)
29 \( 1 + (-0.675 + 2.07i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (8.18 - 5.94i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.11 + 3.43i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.45 - 10.6i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.17T + 43T^{2} \)
47 \( 1 + (-1.29 - 3.97i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.76 + 4.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.67 + 8.23i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.11 - 2.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.55T + 67T^{2} \)
71 \( 1 + (1.16 + 0.845i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.05 - 12.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (14.3 - 10.4i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.34 - 6.79i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.87T + 89T^{2} \)
97 \( 1 + (4.62 - 3.36i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.07506679002424272040609608578, −10.22788337347187669682148418464, −9.225780012964512611217046236825, −8.226105976809262264430743165550, −6.81892926293019357647237495176, −6.14547288449986196067186254221, −5.43778037831583525539011391476, −3.83118178540950920374522978699, −2.72202602981747947463494175394, −1.41170084791829956646693722381, 1.73753360116089454375486257347, 3.68910344414662785944138905605, 4.37167739737226637882980253298, 5.58998766448610914857797864442, 6.30577695060055915489911474259, 7.27838600128063090797153762516, 8.843968273913171154540949192339, 9.132566309346361376486477769390, 10.31709982418338105993139005392, 11.29318010763014506216721611073

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