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Properties

Label 2-546-91.30-c1-0-13
Degree $2$
Conductor $546$
Sign $0.154 + 0.988i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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 + 3-s + (0.499 − 0.866i)4-s + (−0.440 − 0.254i)5-s + (−0.866 + 0.5i)6-s + (−2.39 − 1.13i)7-s + 0.999i·8-s + 9-s + 0.508·10-s − 3.86i·11-s + (0.499 − 0.866i)12-s + (−3.12 + 1.79i)13-s + (2.63 − 0.212i)14-s + (−0.440 − 0.254i)15-s + (−0.5 − 0.866i)16-s + (3.89 − 6.74i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + 0.577·3-s + (0.249 − 0.433i)4-s + (−0.196 − 0.113i)5-s + (−0.353 + 0.204i)6-s + (−0.903 − 0.428i)7-s + 0.353i·8-s + 0.333·9-s + 0.160·10-s − 1.16i·11-s + (0.144 − 0.249i)12-s + (−0.866 + 0.498i)13-s + (0.704 − 0.0569i)14-s + (−0.113 − 0.0656i)15-s + (−0.125 − 0.216i)16-s + (0.944 − 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.154 + 0.988i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.154 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663571 - 0.567914i\)
\(L(\frac12)\) \(\approx\) \(0.663571 - 0.567914i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (2.39 + 1.13i)T \)
13 \( 1 + (3.12 - 1.79i)T \)
good5 \( 1 + (0.440 + 0.254i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.86iT - 11T^{2} \)
17 \( 1 + (-3.89 + 6.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 0.115iT - 19T^{2} \)
23 \( 1 + (0.614 + 1.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.86 + 3.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.83 - 2.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.50 + 0.868i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.138 - 0.0796i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.153 + 0.266i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.61 + 3.81i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.08 + 3.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.24 - 4.18i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.14T + 61T^{2} \)
67 \( 1 + 1.54iT - 67T^{2} \)
71 \( 1 + (0.505 - 0.292i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.78 + 2.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.16 + 3.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.96iT - 83T^{2} \)
89 \( 1 + (-3.92 + 2.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.4 - 7.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.20958127874249256327873909876, −9.672470008876680254088145716212, −8.877673485625406151001816858599, −7.914572908029413345764387129672, −7.16895787884537001693636220127, −6.27410324953145086961793917273, −5.04881265312712607515409768771, −3.64913358187136687301860639530, −2.57783755941505514326384022264, −0.56226146992805151587288014232, 1.82219237260486197244284590270, 3.01727118134753382654965866298, 3.96610109691018684635383360871, 5.48580158802759480569034367588, 6.75180296877054132828134634207, 7.60988070973448404953299562320, 8.346353862247161905195114714998, 9.553084822940824718620553774808, 9.823514468022465279645280122383, 10.73368379029603339809933481865

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