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Properties

Label 2-546-273.251-c1-0-33
Degree $2$
Conductor $546$
Sign $-0.986 - 0.164i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 1.65i)3-s + (−0.499 − 0.866i)4-s − 2.52i·5-s + (−1.68 − 0.396i)6-s + (0.5 − 2.59i)7-s − 0.999·8-s + (−2.5 + 1.65i)9-s + (−2.18 − 1.26i)10-s + (−0.686 + 1.18i)11-s + (−1.18 + 1.26i)12-s + (3.5 − 0.866i)13-s + (−2 − 1.73i)14-s + (−4.18 + 1.26i)15-s + (−0.5 + 0.866i)16-s + (−0.686 − 1.18i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.957i)3-s + (−0.249 − 0.433i)4-s − 1.12i·5-s + (−0.688 − 0.161i)6-s + (0.188 − 0.981i)7-s − 0.353·8-s + (−0.833 + 0.552i)9-s + (−0.691 − 0.399i)10-s + (−0.206 + 0.358i)11-s + (−0.342 + 0.364i)12-s + (0.970 − 0.240i)13-s + (−0.534 − 0.462i)14-s + (−1.08 + 0.325i)15-s + (−0.125 + 0.216i)16-s + (−0.166 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.986 - 0.164i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.986 - 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115439 + 1.39190i\)
\(L(\frac12)\) \(\approx\) \(0.115439 + 1.39190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 1.65i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 + 2.52iT - 5T^{2} \)
11 \( 1 + (0.686 - 1.18i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.686 + 1.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.68 - 2.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.813 - 0.469i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.686 + 0.396i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 + (-7.11 - 4.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.31 + 3.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.25iT - 47T^{2} \)
53 \( 1 + 14.3iT - 53T^{2} \)
59 \( 1 + (8.18 - 4.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.11 + 4.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.558 - 0.967i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.744T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 - 5.04iT - 83T^{2} \)
89 \( 1 + (-10.8 - 6.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.37 - 9.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.60244576020437907831848559120, −9.526868432725057127673113284332, −8.460645260687130113436260318640, −7.73334733343092793009558938534, −6.62438171519103082851791572383, −5.51512702831221887789329282710, −4.69697389091039334831182744701, −3.48519190233748406800961158027, −1.76812145425410242318035840060, −0.78724608820131961066438979290, 2.72042693565101944041330524495, 3.61527386605545495693374086724, 4.81948979405628088833365284234, 5.86919118152622788792982332433, 6.37424947578286834094102646723, 7.61248548259118218785163035858, 8.771190977319440843787651101084, 9.306369205035177067645191949098, 10.61749443232423682967813511515, 11.12760458099002327497787079376

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