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Properties

Label 2-546-91.25-c1-0-7
Degree $2$
Conductor $546$
Sign $0.921 - 0.389i$
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 + (1.95 − 1.13i)5-s − 0.999i·6-s + (2.41 + 1.07i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−1.13 + 1.95i)10-s + (1.09 + 0.630i)11-s + (0.499 + 0.866i)12-s + (−1.26 − 3.37i)13-s + (−2.63 + 0.279i)14-s + 2.26i·15-s + (−0.5 − 0.866i)16-s + (−0.188 + 0.326i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (0.875 − 0.505i)5-s − 0.408i·6-s + (0.914 + 0.405i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.357 + 0.619i)10-s + (0.329 + 0.190i)11-s + (0.144 + 0.249i)12-s + (−0.349 − 0.936i)13-s + (−0.703 + 0.0748i)14-s + 0.583i·15-s + (−0.125 − 0.216i)16-s + (−0.0457 + 0.0793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.921 - 0.389i$
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: \((2,\ 546,\ (\ :1/2),\ 0.921 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25453 + 0.254408i\)
\(L(\frac12)\) \(\approx\) \(1.25453 + 0.254408i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 1.07i)T \)
13 \( 1 + (1.26 + 3.37i)T \)
good5 \( 1 + (-1.95 + 1.13i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.09 - 0.630i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.188 - 0.326i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.71 + 3.87i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.24 + 3.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.40T + 29T^{2} \)
31 \( 1 + (-3.46 - 2i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.92 + 1.68i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.90iT - 41T^{2} \)
43 \( 1 - 8.75T + 43T^{2} \)
47 \( 1 + (2.85 - 1.64i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.04 + 2.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.45 - 7.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.11 - 0.646i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.11iT - 71T^{2} \)
73 \( 1 + (8.26 + 4.77i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.14 - 5.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.39iT - 83T^{2} \)
89 \( 1 + (-9.85 + 5.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.03iT - 97T^{2} \)
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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

−10.71575278429201629104852476501, −9.736071707255483048069969793985, −9.272201955094791916219800429522, −8.300912937644418994847649635468, −7.41738586374434630575983759676, −6.09649395139263325883047287462, −5.35837095233621081221624887591, −4.61311251569959640474744895914, −2.68919927081410101925914737569, −1.19205094125529158094678193076, 1.34910450394577371279894825807, 2.29980101764674622600741329056, 3.86560591269196317593686573709, 5.31907091565483133497947190730, 6.29892367882582921114024266900, 7.31263287454439653356046357179, 7.927805573440275529259208318898, 9.182504356466658141547017133668, 9.888690745218809514101768732953, 10.72629851938139032985726208869

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