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Properties

Label 2-546-91.31-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.853 - 0.520i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)3-s + (0.866 + 0.499i)4-s + (−0.0458 + 0.171i)5-s + (−0.707 − 0.707i)6-s + (−2.36 − 1.18i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.0885 − 0.153i)10-s + (−5.25 + 1.40i)11-s + (0.5 + 0.866i)12-s + (−2.49 + 2.60i)13-s + (1.97 + 1.76i)14-s + (−0.125 + 0.125i)15-s + (0.500 + 0.866i)16-s + (−0.452 + 0.784i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.499 + 0.288i)3-s + (0.433 + 0.249i)4-s + (−0.0205 + 0.0765i)5-s + (−0.288 − 0.288i)6-s + (−0.893 − 0.449i)7-s + (−0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.0280 − 0.0485i)10-s + (−1.58 + 0.424i)11-s + (0.144 + 0.249i)12-s + (−0.691 + 0.722i)13-s + (0.527 + 0.470i)14-s + (−0.0323 + 0.0323i)15-s + (0.125 + 0.216i)16-s + (−0.109 + 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0935048 + 0.333013i\)
\(L(\frac12)\) \(\approx\) \(0.0935048 + 0.333013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (2.36 + 1.18i)T \)
13 \( 1 + (2.49 - 2.60i)T \)
good5 \( 1 + (0.0458 - 0.171i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (5.25 - 1.40i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.452 - 0.784i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0556 + 0.207i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (7.19 - 4.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.101T + 29T^{2} \)
31 \( 1 + (-0.893 + 0.239i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.12 + 4.21i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.40 + 3.40i)T + 41iT^{2} \)
43 \( 1 + 3.97iT - 43T^{2} \)
47 \( 1 + (4.67 + 1.25i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.98 - 6.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.32 - 8.69i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.115 + 0.0668i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.20 - 8.24i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.61 + 3.61i)T - 71iT^{2} \)
73 \( 1 + (-1.61 - 6.04i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.25 + 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.96 + 6.96i)T + 83iT^{2} \)
89 \( 1 + (6.53 + 1.75i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.96 + 7.96i)T + 97iT^{2} \)
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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

−10.79722858244632092628061985814, −10.10782869440487353019144374316, −9.602503714225419415869140759643, −8.613667701701582404929165421708, −7.58093456243787864716124329547, −7.05487094501328522268180300832, −5.68910553433098547289804671306, −4.35919082078756371757328430091, −3.16626602766577372639063360625, −2.11995326789086078885559573452, 0.21128405665226217511339368051, 2.38483198580350654207500623320, 3.12385173408608688634530816119, 4.95257088744620063256802511092, 6.03564476211045369335815378529, 6.92281681384699696018035499873, 8.183296694467144343118389805722, 8.274504553253554633679365300006, 9.736679968721496383893317663062, 10.06071219282885798045575404805

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