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Properties

Label 2-546-91.89-c1-0-7
Degree $2$
Conductor $546$
Sign $0.978 + 0.207i$
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 + (0.0344 + 0.128i)5-s + (0.258 + 0.965i)6-s + (2.22 − 1.42i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.0666 − 0.115i)10-s + (1.12 + 4.19i)11-s + (0.500 − 0.866i)12-s + (−1.49 + 3.28i)13-s + (−2.58 − 0.569i)14-s + (0.0344 − 0.128i)15-s − 1.00·16-s − 3.34·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.499 − 0.288i)3-s + 0.500i·4-s + (0.0154 + 0.0575i)5-s + (0.105 + 0.394i)6-s + (0.842 − 0.538i)7-s + (0.250 − 0.250i)8-s + (0.166 + 0.288i)9-s + (0.0210 − 0.0364i)10-s + (0.338 + 1.26i)11-s + (0.144 − 0.250i)12-s + (−0.415 + 0.909i)13-s + (−0.690 − 0.152i)14-s + (0.00890 − 0.0332i)15-s − 0.250·16-s − 0.810·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02649 - 0.107856i\)
\(L(\frac12)\) \(\approx\) \(1.02649 - 0.107856i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 1.42i)T \)
13 \( 1 + (1.49 - 3.28i)T \)
good5 \( 1 + (-0.0344 - 0.128i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.12 - 4.19i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 3.34T + 17T^{2} \)
19 \( 1 + (-3.90 - 1.04i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 + (-2.15 - 3.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.57 - 0.690i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-5.09 + 5.09i)T - 37iT^{2} \)
41 \( 1 + (-11.0 - 2.96i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.78 - 5.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (11.3 - 3.04i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.41 + 5.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.99 + 1.99i)T + 59iT^{2} \)
61 \( 1 + (-5.17 + 2.99i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.21 - 2.20i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-12.0 + 3.24i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.11 + 7.88i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.41 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.93 - 8.93i)T - 83iT^{2} \)
89 \( 1 + (7.78 + 7.78i)T + 89iT^{2} \)
97 \( 1 + (1.45 + 5.41i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.96879778020581184995033441330, −9.907438029052595209527300514387, −9.223084046668591424472688857405, −8.019787078261111766136521672310, −7.23968354874355299307442195663, −6.49888000977321823946885094404, −4.83366004558220343702170543471, −4.27222409692636804734533903912, −2.41547036428264061954428682471, −1.25569065328831456487173201064, 0.929029175082324716526583153791, 2.82381466672646786068168581138, 4.47845452504052955955460726153, 5.46417692970199165364588341282, 6.08939914925257117206010914599, 7.32463205478743377236711541625, 8.241174401275361395067206682511, 8.978851368491522527739283859900, 9.847687018226717939992336529380, 11.01313032500976418949235929126

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