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Properties

Label 2-546-273.32-c1-0-34
Degree $2$
Conductor $546$
Sign $-0.891 - 0.452i$
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 + (−0.315 − 1.70i)3-s + 1.00i·4-s + (0.258 − 0.965i)5-s + (−0.981 + 1.42i)6-s + (−0.443 − 2.60i)7-s + (0.707 − 0.707i)8-s + (−2.80 + 1.07i)9-s + (−0.865 + 0.499i)10-s + (−2.62 − 0.704i)11-s + (1.70 − 0.315i)12-s + (3.59 − 0.238i)13-s + (−1.53 + 2.15i)14-s + (−1.72 − 0.136i)15-s − 1.00·16-s − 2.39·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.182 − 0.983i)3-s + 0.500i·4-s + (0.115 − 0.431i)5-s + (−0.400 + 0.582i)6-s + (−0.167 − 0.985i)7-s + (0.250 − 0.250i)8-s + (−0.933 + 0.357i)9-s + (−0.273 + 0.158i)10-s + (−0.792 − 0.212i)11-s + (0.491 − 0.0910i)12-s + (0.997 − 0.0661i)13-s + (−0.409 + 0.576i)14-s + (−0.445 − 0.0351i)15-s − 0.250·16-s − 0.580·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.891 - 0.452i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.891 - 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146315 + 0.611537i\)
\(L(\frac12)\) \(\approx\) \(0.146315 + 0.611537i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.315 + 1.70i)T \)
7 \( 1 + (0.443 + 2.60i)T \)
13 \( 1 + (-3.59 + 0.238i)T \)
good5 \( 1 + (-0.258 + 0.965i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.62 + 0.704i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 + (0.807 + 3.01i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.02T + 23T^{2} \)
29 \( 1 + (2.49 + 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.354 + 1.32i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.770 - 0.770i)T + 37iT^{2} \)
41 \( 1 + (-0.406 + 0.108i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.05 - 1.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.717 + 0.192i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.12 - 1.22i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.69 - 7.69i)T - 59iT^{2} \)
61 \( 1 + (0.903 - 1.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.116 + 0.0311i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.68 + 13.7i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.19 + 1.12i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.71 + 9.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.50 - 7.50i)T + 83iT^{2} \)
89 \( 1 + (-7.22 + 7.22i)T - 89iT^{2} \)
97 \( 1 + (4.73 + 1.26i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.58593273192589392273118346881, −9.353578476922765623478424329550, −8.423841339757570582679783765163, −7.74540996037246625930003918103, −6.81609652906684209842035003616, −5.84136600736748458840428955279, −4.47804738720902255582868604014, −3.12402180251519430286391874536, −1.73837403745179431476580199810, −0.42551733841478773088251845829, 2.31288397006085053714716797173, 3.65751505066157277789853752089, 4.98586050842772735836083381680, 5.88149261406093591061210498867, 6.54290969994329234545311564145, 8.038685600386251117761565011479, 8.686169461121446231831675391023, 9.542738084282700923710077593202, 10.35407041690237269126354935612, 10.97495893373480264335346313667

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