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Properties

Label 2-546-91.47-c1-0-6
Degree $2$
Conductor $546$
Sign $0.794 - 0.607i$
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 + (0.799 + 2.98i)5-s + (−0.707 + 0.707i)6-s + (−1.93 − 1.80i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.54 − 2.67i)10-s + (2.46 + 0.659i)11-s + (0.5 − 0.866i)12-s + (3.14 − 1.77i)13-s + (2.33 + 1.23i)14-s + (2.18 + 2.18i)15-s + (0.500 − 0.866i)16-s + (0.0961 + 0.166i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (0.357 + 1.33i)5-s + (−0.288 + 0.288i)6-s + (−0.732 − 0.681i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.488 − 0.846i)10-s + (0.742 + 0.198i)11-s + (0.144 − 0.249i)12-s + (0.871 − 0.491i)13-s + (0.624 + 0.331i)14-s + (0.564 + 0.564i)15-s + (0.125 − 0.216i)16-s + (0.0233 + 0.0403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25935 + 0.425971i\)
\(L(\frac12)\) \(\approx\) \(1.25935 + 0.425971i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 + 1.80i)T \)
13 \( 1 + (-3.14 + 1.77i)T \)
good5 \( 1 + (-0.799 - 2.98i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.46 - 0.659i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.0961 - 0.166i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.54 - 5.76i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.28 - 3.05i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.72T + 29T^{2} \)
31 \( 1 + (4.78 + 1.28i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.22 - 8.30i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.08 - 6.08i)T - 41iT^{2} \)
43 \( 1 + 8.17iT - 43T^{2} \)
47 \( 1 + (-11.2 + 3.02i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.33 - 4.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.23 + 8.34i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.60 + 5.54i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.28 + 4.80i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.54 - 3.54i)T + 71iT^{2} \)
73 \( 1 + (2.53 - 9.47i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.52 + 6.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.125 - 0.125i)T - 83iT^{2} \)
89 \( 1 + (13.8 - 3.70i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.26 + 7.26i)T - 97iT^{2} \)
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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

−10.60577517069285024360635954633, −10.01868427979866950701647640432, −9.245649160674475557766136053973, −8.147114709746143502318455761049, −7.19150703218671589784965764844, −6.65958321671979328579390625062, −5.81294404346681557973172673397, −3.71879948716512461637231900723, −3.00614831741187924914227142845, −1.43670829421365037045670393160, 1.08497804483207523829196887400, 2.54965682257197916973980161268, 3.84932428636832255424321078348, 5.07388068284101204370575959664, 6.17764825485769470975996606268, 7.22202607719483480151743139701, 8.656315317091092356401872188431, 9.063240407123679939249227511691, 9.281016949290663753216100978128, 10.56953285994367495325643675006

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