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Properties

Label 2-546-91.59-c1-0-11
Degree $2$
Conductor $546$
Sign $0.851 + 0.523i$
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  + (−0.707 − 0.707i)2-s + (0.866 − 0.5i)3-s + 1.00i·4-s + (2.92 + 0.782i)5-s + (−0.965 − 0.258i)6-s + (2.42 − 1.04i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.51 − 2.61i)10-s + (2.12 + 0.570i)11-s + (0.500 + 0.866i)12-s + (−1.15 + 3.41i)13-s + (−2.45 − 0.977i)14-s + (2.92 − 0.782i)15-s − 1.00·16-s − 4.80·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.499 − 0.288i)3-s + 0.500i·4-s + (1.30 + 0.349i)5-s + (−0.394 − 0.105i)6-s + (0.918 − 0.395i)7-s + (0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.478 − 0.828i)10-s + (0.641 + 0.172i)11-s + (0.144 + 0.250i)12-s + (−0.319 + 0.947i)13-s + (−0.657 − 0.261i)14-s + (0.754 − 0.202i)15-s − 0.250·16-s − 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.851 + 0.523i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.851 + 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72018 - 0.486667i\)
\(L(\frac12)\) \(\approx\) \(1.72018 - 0.486667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.42 + 1.04i)T \)
13 \( 1 + (1.15 - 3.41i)T \)
good5 \( 1 + (-2.92 - 0.782i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.12 - 0.570i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 4.80T + 17T^{2} \)
19 \( 1 + (-0.738 - 2.75i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.832iT - 23T^{2} \)
29 \( 1 + (-0.310 + 0.538i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.547 - 2.04i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.91 - 2.91i)T - 37iT^{2} \)
41 \( 1 + (2.82 + 10.5i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.69 - 3.28i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.02 + 3.82i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.61 + 9.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.57 + 5.57i)T + 59iT^{2} \)
61 \( 1 + (6.58 + 3.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.10 - 7.84i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.25 + 8.42i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-15.5 + 4.17i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.859 - 1.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.5 - 12.5i)T - 83iT^{2} \)
89 \( 1 + (5.24 + 5.24i)T + 89iT^{2} \)
97 \( 1 + (11.7 + 3.14i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.57606525802449054589351060076, −9.821117459381175310623626482006, −9.052955889034659991979122702622, −8.305054984441066837391886978642, −7.08250290848449984870936313678, −6.47732649567969186427199818024, −4.98454193148994859521476461039, −3.79984058262290169170478759409, −2.21576071495235304969305756983, −1.64656728990944896613479962773, 1.51267849361985569178313988571, 2.64500171635987484680917828268, 4.53657338019000699826617840168, 5.38080673006912008425970450192, 6.25431273432464293212120042995, 7.41982553970242607364437932382, 8.480028329239822885546220379905, 9.031672828434378061130803924219, 9.745326318412188928542354175823, 10.63029901890607436363225819543

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