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Properties

Label 2-546-91.45-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: $\chi_{546} (409, \cdot )$
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

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

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