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Properties

Label 2-462-231.65-c1-0-19
Degree $2$
Conductor $462$
Sign $-0.394 + 0.918i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (−0.5 + 0.866i)2-s + (−1.20 − 1.24i)3-s + (−0.499 − 0.866i)4-s + (−1.00 − 0.580i)5-s + (1.67 − 0.426i)6-s + (2.19 − 1.47i)7-s + 0.999·8-s + (−0.0797 + 2.99i)9-s + (1.00 − 0.580i)10-s + (0.864 + 3.20i)11-s + (−0.470 + 1.66i)12-s − 3.56i·13-s + (0.173 + 2.64i)14-s + (0.494 + 1.94i)15-s + (−0.5 + 0.866i)16-s + (−1.71 − 2.96i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.697 − 0.716i)3-s + (−0.249 − 0.433i)4-s + (−0.449 − 0.259i)5-s + (0.685 − 0.173i)6-s + (0.831 − 0.555i)7-s + 0.353·8-s + (−0.0265 + 0.999i)9-s + (0.318 − 0.183i)10-s + (0.260 + 0.965i)11-s + (−0.135 + 0.481i)12-s − 0.989i·13-s + (0.0463 + 0.705i)14-s + (0.127 + 0.503i)15-s + (−0.125 + 0.216i)16-s + (−0.415 − 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306311 - 0.465029i\)
\(L(\frac12)\) \(\approx\) \(0.306311 - 0.465029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (1.20 + 1.24i)T \)
7 \( 1 + (-2.19 + 1.47i)T \)
11 \( 1 + (-0.864 - 3.20i)T \)
good5 \( 1 + (1.00 + 0.580i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.56iT - 13T^{2} \)
17 \( 1 + (1.71 + 2.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.30 + 2.48i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.67 + 4.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.04T + 29T^{2} \)
31 \( 1 + (0.547 + 0.947i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.31 - 9.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.05T + 41T^{2} \)
43 \( 1 + 6.97iT - 43T^{2} \)
47 \( 1 + (4.23 + 2.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.75 + 1.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.87 - 5.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.96 - 5.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.911 + 1.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-8.95 + 5.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.06 - 4.65i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.21T + 83T^{2} \)
89 \( 1 + (2.99 + 1.72i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.20T + 97T^{2} \)
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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

−10.61160856745617255081116954547, −10.12435474632610956292212896462, −8.480131209359819833124090532667, −8.022507108068166485041214754127, −7.07154378905408852443526918795, −6.35425579507205688275418792760, −5.00432802169360250372910062114, −4.40609928262840295467012366219, −2.03809010488631240399017613786, −0.42109799182226468009848994810, 1.80754412047241851610936550885, 3.60720573413042954322495411096, 4.32317842650423849542964206473, 5.61256829549711099025545866092, 6.55171470223429002785522107844, 8.046149102696728209604553425288, 8.733777574902726577159350343080, 9.661891054961258736323007687929, 10.68630350149944522562314331763, 11.29124418790827629559351924846

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