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Properties

Label 2-546-273.269-c1-0-10
Degree $2$
Conductor $546$
Sign $-0.871 - 0.491i$
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  + i·2-s + (1.38 + 1.04i)3-s − 4-s + (1.07 + 1.86i)5-s + (−1.04 + 1.38i)6-s + (−2.48 − 0.916i)7-s − i·8-s + (0.812 + 2.88i)9-s + (−1.86 + 1.07i)10-s + (−0.720 + 0.416i)11-s + (−1.38 − 1.04i)12-s + (−1.46 + 3.29i)13-s + (0.916 − 2.48i)14-s + (−0.463 + 3.69i)15-s + 16-s − 2.72·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.797 + 0.603i)3-s − 0.5·4-s + (0.480 + 0.832i)5-s + (−0.426 + 0.563i)6-s + (−0.938 − 0.346i)7-s − 0.353i·8-s + (0.270 + 0.962i)9-s + (−0.588 + 0.339i)10-s + (−0.217 + 0.125i)11-s + (−0.398 − 0.301i)12-s + (−0.405 + 0.914i)13-s + (0.244 − 0.663i)14-s + (−0.119 + 0.954i)15-s + 0.250·16-s − 0.660·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.871 - 0.491i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.871 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401305 + 1.52911i\)
\(L(\frac12)\) \(\approx\) \(0.401305 + 1.52911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.38 - 1.04i)T \)
7 \( 1 + (2.48 + 0.916i)T \)
13 \( 1 + (1.46 - 3.29i)T \)
good5 \( 1 + (-1.07 - 1.86i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.720 - 0.416i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.72T + 17T^{2} \)
19 \( 1 + (-2.22 - 1.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.48iT - 23T^{2} \)
29 \( 1 + (-7.62 - 4.40i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.06 + 1.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.179T + 37T^{2} \)
41 \( 1 + (-5.93 + 10.2i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.89 + 6.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.425 - 0.736i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 5.99i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.52T + 59T^{2} \)
61 \( 1 + (-3.16 - 1.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.30 - 5.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.48 + 4.32i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (10.9 + 6.31i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.10 + 5.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.930T + 83T^{2} \)
89 \( 1 + 1.14T + 89T^{2} \)
97 \( 1 + (7.08 - 4.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.68313496502797483621305753130, −10.15999860541228518275716009657, −9.350110915742616761725671773676, −8.677707009399554713924680358590, −7.31854585552776379352196558697, −6.89421292151493928948018910889, −5.74499432237181879918305976965, −4.49851694939986868060454300807, −3.48105335970635779101768744745, −2.37419286941287282920701745680, 0.844146518521171508867127660562, 2.40687825772940834573020013427, 3.17647406006019123659988930672, 4.60117464003240353214536440069, 5.77115504397092234524955284622, 6.81590018920310557720151358275, 8.087139416009143510625979017745, 8.757959788628922600670655539654, 9.594035440922415659904782174366, 10.10853776651248778381102444765

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