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Properties

Label 2-546-91.59-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.0262 - 0.999i$
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 + (2.65 + 0.711i)5-s + (−0.965 − 0.258i)6-s + (1.40 + 2.24i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.37 + 2.37i)10-s + (2.14 + 0.573i)11-s + (−0.500 − 0.866i)12-s + (−0.354 − 3.58i)13-s + (−0.589 + 2.57i)14-s + (−2.65 + 0.711i)15-s − 1.00·16-s − 1.67·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.499 + 0.288i)3-s + 0.500i·4-s + (1.18 + 0.318i)5-s + (−0.394 − 0.105i)6-s + (0.531 + 0.846i)7-s + (−0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (0.434 + 0.752i)10-s + (0.645 + 0.173i)11-s + (−0.144 − 0.250i)12-s + (−0.0983 − 0.995i)13-s + (−0.157 + 0.689i)14-s + (−0.685 + 0.183i)15-s − 0.250·16-s − 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0262 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0262 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0262 - 0.999i$
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: \((2,\ 546,\ (\ :1/2),\ -0.0262 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41687 + 1.45464i\)
\(L(\frac12)\) \(\approx\) \(1.41687 + 1.45464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 2.24i)T \)
13 \( 1 + (0.354 + 3.58i)T \)
good5 \( 1 + (-2.65 - 0.711i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.14 - 0.573i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 + (0.0474 + 0.177i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.546iT - 23T^{2} \)
29 \( 1 + (2.18 - 3.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.02 - 7.55i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.909 + 0.909i)T - 37iT^{2} \)
41 \( 1 + (1.32 + 4.95i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.86 - 1.65i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.61 + 9.74i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.78 + 3.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.05 + 1.05i)T + 59iT^{2} \)
61 \( 1 + (11.8 + 6.83i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.82 - 6.80i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.04 + 3.91i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.57 + 2.29i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.42 - 7.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.3 + 11.3i)T - 83iT^{2} \)
89 \( 1 + (11.3 + 11.3i)T + 89iT^{2} \)
97 \( 1 + (-15.6 - 4.18i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.00830818974324403771481232373, −10.19341433574600232589703367261, −9.234651724618169203274333139822, −8.440326358881167898768167296923, −7.12202710757183908644496256235, −6.22478774510272880245936420074, −5.50897986929195706817688452096, −4.81064263413041615188468331278, −3.28102577499167957704164020189, −1.93302234076316023211919565030, 1.22718426988988382984937416475, 2.21711623385934761832262289492, 4.05429005120371987499535853568, 4.82636963893958823080504973969, 5.98327930937716232603419854547, 6.58890833296800172571050169418, 7.78253051886656796312485851723, 9.169085266916246557833525885829, 9.771954793422990476811956446722, 10.76768129783635065135468553034

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