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Properties

Label 2-546-21.5-c1-0-26
Degree $2$
Conductor $546$
Sign $-0.268 + 0.963i$
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  + (−0.866 − 0.5i)2-s + (1.11 − 1.32i)3-s + (0.499 + 0.866i)4-s + (0.737 − 1.27i)5-s + (−1.62 + 0.591i)6-s + (2.24 − 1.39i)7-s − 0.999i·8-s + (−0.517 − 2.95i)9-s + (−1.27 + 0.737i)10-s + (0.616 − 0.356i)11-s + (1.70 + 0.301i)12-s + i·13-s + (−2.64 + 0.0805i)14-s + (−0.872 − 2.40i)15-s + (−0.5 + 0.866i)16-s + (0.629 + 1.09i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.643 − 0.765i)3-s + (0.249 + 0.433i)4-s + (0.329 − 0.571i)5-s + (−0.664 + 0.241i)6-s + (0.850 − 0.526i)7-s − 0.353i·8-s + (−0.172 − 0.984i)9-s + (−0.404 + 0.233i)10-s + (0.186 − 0.107i)11-s + (0.492 + 0.0870i)12-s + 0.277i·13-s + (−0.706 + 0.0215i)14-s + (−0.225 − 0.620i)15-s + (−0.125 + 0.216i)16-s + (0.152 + 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.268 + 0.963i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913814 - 1.20274i\)
\(L(\frac12)\) \(\approx\) \(0.913814 - 1.20274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.11 + 1.32i)T \)
7 \( 1 + (-2.24 + 1.39i)T \)
13 \( 1 - iT \)
good5 \( 1 + (-0.737 + 1.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.616 + 0.356i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.629 - 1.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.83 + 1.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.87 - 1.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.97iT - 29T^{2} \)
31 \( 1 + (1.42 - 0.824i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.738 + 1.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.74T + 41T^{2} \)
43 \( 1 + 8.19T + 43T^{2} \)
47 \( 1 + (-1.02 + 1.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.4 - 6.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.88 - 8.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.61 + 2.08i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.50 + 6.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.81iT - 71T^{2} \)
73 \( 1 + (1.31 - 0.760i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.00 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + (-1.87 + 3.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.98iT - 97T^{2} \)
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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

−10.54554802326411455687479603370, −9.380990376637862595240627365491, −8.822541845768557667489171076008, −7.961032374037453247529693934480, −7.25760314641934997718906503539, −6.18599278115850000102655356422, −4.75656677885824109373225273134, −3.49783460766130639525657360598, −2.05180604836957559433878513556, −1.09702911951868143355183528174, 1.94477230526401543301509313693, 3.05316998860173171005174296700, 4.55827979122301465062890733408, 5.48520362609591937105130360259, 6.62525527434611896024949890007, 7.77431524502299630007010798641, 8.448651364623158189461841481363, 9.253047196614622150526751380790, 10.08067418497695662421035085563, 10.80810444124660427342800150950

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