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Properties

Label 2-546-91.25-c1-0-11
Degree $2$
Conductor $546$
Sign $0.421 + 0.906i$
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  + (0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.294 + 0.169i)5-s + 0.999i·6-s + (−0.420 − 2.61i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.169 + 0.294i)10-s + (0.571 + 0.330i)11-s + (0.499 + 0.866i)12-s + (0.660 − 3.54i)13-s + (−1.66 − 2.05i)14-s − 0.339i·15-s + (−0.5 − 0.866i)16-s + (3.27 − 5.66i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.131 + 0.0759i)5-s + 0.408i·6-s + (−0.158 − 0.987i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.0537 + 0.0930i)10-s + (0.172 + 0.0995i)11-s + (0.144 + 0.249i)12-s + (0.183 − 0.983i)13-s + (−0.446 − 0.548i)14-s − 0.0877i·15-s + (−0.125 − 0.216i)16-s + (0.793 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.421 + 0.906i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.421 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49308 - 0.952000i\)
\(L(\frac12)\) \(\approx\) \(1.49308 - 0.952000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.420 + 2.61i)T \)
13 \( 1 + (-0.660 + 3.54i)T \)
good5 \( 1 + (0.294 - 0.169i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.571 - 0.330i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.27 + 5.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.27 + 3.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.71 - 6.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 + (3.46 + 2i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.06 + 1.77i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.864iT - 41T^{2} \)
43 \( 1 + 5.08T + 43T^{2} \)
47 \( 1 + (-8.18 + 4.72i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.38 - 1.95i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.932 + 1.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.45 + 3.72i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.884iT - 71T^{2} \)
73 \( 1 + (8.72 + 5.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.22 - 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 + (3.85 - 2.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−10.88928208802498643375008602200, −9.768075278874781161885440292210, −9.378105113329569020944402683925, −7.59285060247207162123959125727, −7.14093036225472140285609745677, −5.62406143322400478914506171183, −5.06589899039714186344388593293, −3.74952563644273680221462882028, −3.09307929547889309573054623282, −0.953690556038445207637662664664, 1.77859430730049712702269206016, 3.19865492652873694730471448942, 4.42334769660902651855021487171, 5.66450302335608289111892656773, 6.17435058307518281665355818575, 7.20773966772266240843047305464, 8.214697691240171934168731250575, 8.955067172585349886526442645728, 10.14796600257992164590351609491, 11.29115924436224997290882361695

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