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Properties

Label 2-546-91.45-c1-0-8
Degree $2$
Conductor $546$
Sign $0.959 + 0.283i$
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 + (−0.445 + 1.66i)5-s + (−0.258 + 0.965i)6-s + (−2.60 − 0.449i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.859 − 1.48i)10-s + (1.45 − 5.43i)11-s + (−0.500 − 0.866i)12-s + (1.62 − 3.22i)13-s + (2.16 − 1.52i)14-s + (0.445 + 1.66i)15-s − 1.00·16-s + 7.65·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (−0.199 + 0.742i)5-s + (−0.105 + 0.394i)6-s + (−0.985 − 0.170i)7-s + (0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.271 − 0.471i)10-s + (0.439 − 1.64i)11-s + (−0.144 − 0.250i)12-s + (0.449 − 0.893i)13-s + (0.577 − 0.407i)14-s + (0.114 + 0.428i)15-s − 0.250·16-s + 1.85·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.959 + 0.283i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.959 + 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19229 - 0.172373i\)
\(L(\frac12)\) \(\approx\) \(1.19229 - 0.172373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 + 0.449i)T \)
13 \( 1 + (-1.62 + 3.22i)T \)
good5 \( 1 + (0.445 - 1.66i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.45 + 5.43i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 + (0.350 - 0.0939i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 6.34iT - 23T^{2} \)
29 \( 1 + (-3.90 + 6.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.143 + 0.0384i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.94 + 4.94i)T + 37iT^{2} \)
41 \( 1 + (-8.72 + 2.33i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.41 - 3.70i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.730 - 0.195i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.431 + 0.747i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.917 - 0.917i)T - 59iT^{2} \)
61 \( 1 + (6.74 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.22 - 0.863i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (14.4 + 3.87i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.26 + 8.44i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.10 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.17 + 1.17i)T + 83iT^{2} \)
89 \( 1 + (-2.96 + 2.96i)T - 89iT^{2} \)
97 \( 1 + (0.740 - 2.76i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.55547006283889729448943587023, −9.812412655526450481426619748430, −8.911016378221528897417417102749, −7.967285512342067270261676767797, −7.34415097924079229771630000040, −6.20077641128407466924363717288, −5.71512947589484870878467995077, −3.54040970095836311198826771882, −3.09690751686431929694100080464, −0.903691236067017288176992647533, 1.41845991957231587985413270417, 2.87824312452772368691535130827, 4.01350933060449972947005262172, 4.90521588126169585702571333138, 6.51501991618823312826140865336, 7.38294583438016871934602919126, 8.545948615144603259316160942730, 9.114054752732951435209416750113, 9.931883400559721503428543194835, 10.44568750116271012369911168335

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