Properties

Label 2-546-39.5-c1-0-27
Degree $2$
Conductor $546$
Sign $-0.391 - 0.920i$
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 + (0.273 − 1.71i)3-s + 1.00i·4-s + (−1.75 − 1.75i)5-s + (−1.40 + 1.01i)6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−2.85 − 0.936i)9-s + 2.47i·10-s + (−2.12 + 2.12i)11-s + (1.71 + 0.273i)12-s + (−3.47 + 0.950i)13-s − 1.00i·14-s + (−3.47 + 2.51i)15-s − 1.00·16-s − 3.03·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.158 − 0.987i)3-s + 0.500i·4-s + (−0.783 − 0.783i)5-s + (−0.572 + 0.414i)6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.950 − 0.312i)9-s + 0.783i·10-s + (−0.640 + 0.640i)11-s + (0.493 + 0.0790i)12-s + (−0.964 + 0.263i)13-s − 0.267i·14-s + (−0.897 + 0.649i)15-s − 0.250·16-s − 0.736·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105087 + 0.158863i\)
\(L(\frac12)\) \(\approx\) \(0.105087 + 0.158863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.273 + 1.71i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (3.47 - 0.950i)T \)
good5 \( 1 + (1.75 + 1.75i)T + 5iT^{2} \)
11 \( 1 + (2.12 - 2.12i)T - 11iT^{2} \)
17 \( 1 + 3.03T + 17T^{2} \)
19 \( 1 + (-2.63 + 2.63i)T - 19iT^{2} \)
23 \( 1 + 0.478T + 23T^{2} \)
29 \( 1 - 6.03iT - 29T^{2} \)
31 \( 1 + (2.07 - 2.07i)T - 31iT^{2} \)
37 \( 1 + (4.21 + 4.21i)T + 37iT^{2} \)
41 \( 1 + (4.84 + 4.84i)T + 41iT^{2} \)
43 \( 1 - 4.59iT - 43T^{2} \)
47 \( 1 + (-6.97 + 6.97i)T - 47iT^{2} \)
53 \( 1 - 5.90iT - 53T^{2} \)
59 \( 1 + (3.76 - 3.76i)T - 59iT^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + (6.61 - 6.61i)T - 67iT^{2} \)
71 \( 1 + (7.36 + 7.36i)T + 71iT^{2} \)
73 \( 1 + (8.06 + 8.06i)T + 73iT^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (9.96 + 9.96i)T + 83iT^{2} \)
89 \( 1 + (10.9 - 10.9i)T - 89iT^{2} \)
97 \( 1 + (-10.7 + 10.7i)T - 97iT^{2} \)
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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

−10.22078168421156261646378388225, −8.944488588276728890389499238200, −8.622301844719131456815808029877, −7.42177073899908464674963451450, −7.13983215224669884595489122276, −5.40439539714285736385231512829, −4.42848379866127542606692488823, −2.87891035241974674842559307779, −1.75783261071597281411913537212, −0.12103646265725557721954828456, 2.65312522951132889475302429429, 3.78033953537392025136383081999, 4.89360136182073225413467256752, 5.87458511412314393376041845525, 7.16847617800777413565850071307, 7.898324595826882936460154835747, 8.630736412944401288769331389365, 9.811505273033259520797316203871, 10.34689651347223440074877855818, 11.21389494500725843433155087459

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