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Properties

Label 2-546-273.38-c1-0-15
Degree $2$
Conductor $546$
Sign $0.347 + 0.937i$
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 + (−1.31 − 1.13i)3-s + (−0.499 + 0.866i)4-s + (−1.08 + 0.627i)5-s + (−0.326 + 1.70i)6-s + (−1.14 + 2.38i)7-s + 0.999·8-s + (0.432 + 2.96i)9-s + (1.08 + 0.627i)10-s + (0.900 − 1.56i)11-s + (1.63 − 0.568i)12-s + (0.657 − 3.54i)13-s + (2.63 − 0.204i)14-s + (2.13 + 0.409i)15-s + (−0.5 − 0.866i)16-s + (1.12 − 1.95i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.756 − 0.654i)3-s + (−0.249 + 0.433i)4-s + (−0.486 + 0.280i)5-s + (−0.133 + 0.694i)6-s + (−0.431 + 0.902i)7-s + 0.353·8-s + (0.144 + 0.989i)9-s + (0.343 + 0.198i)10-s + (0.271 − 0.470i)11-s + (0.472 − 0.164i)12-s + (0.182 − 0.983i)13-s + (0.704 − 0.0545i)14-s + (0.551 + 0.105i)15-s + (−0.125 − 0.216i)16-s + (0.273 − 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.347 + 0.937i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.347 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632945 - 0.440498i\)
\(L(\frac12)\) \(\approx\) \(0.632945 - 0.440498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 1.13i)T \)
7 \( 1 + (1.14 - 2.38i)T \)
13 \( 1 + (-0.657 + 3.54i)T \)
good5 \( 1 + (1.08 - 0.627i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.900 + 1.56i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.12 + 1.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.46 - 4.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.93 + 4.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.62iT - 29T^{2} \)
31 \( 1 + (2.39 - 4.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.77 + 2.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.52iT - 41T^{2} \)
43 \( 1 - 3.14T + 43T^{2} \)
47 \( 1 + (8.20 - 4.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.2 - 6.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.80 - 2.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.64 + 1.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.4 - 6.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 + (-7.95 + 13.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.323 + 0.560i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + (11.1 - 6.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.48T + 97T^{2} \)
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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

−10.86736198698210290249143740738, −9.916588562748716200847455848240, −8.861155304189627943809677039673, −7.960015144692574937878835799694, −7.12517588056312780323152854102, −5.97955949352512501196870893016, −5.19416331623109618454006166179, −3.56440081714085091727180359396, −2.50208316586576874765985815025, −0.78669090679710897094653861443, 0.980349162455806046233455625954, 3.59459238803883114395645534315, 4.48013039115657103082951287855, 5.34225365933690537862792556652, 6.72209423245889877663564340424, 7.03041503282267641375790378274, 8.340444096696143276348380796060, 9.492860379988163890843684245690, 9.776130579504249312202947908603, 11.09529588683753967845266203383

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