Properties

Label 2-462-77.58-c1-0-2
Degree $2$
Conductor $462$
Sign $-0.346 - 0.938i$
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  + (−0.104 − 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (0.363 − 0.161i)5-s + (0.809 + 0.587i)6-s + (−1.41 + 2.23i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + (−0.198 − 0.344i)10-s + (−0.820 − 3.21i)11-s + (0.5 − 0.866i)12-s + (−2.60 + 1.89i)13-s + (2.37 + 1.17i)14-s + (−0.122 + 0.378i)15-s + (0.913 − 0.406i)16-s + (−0.654 + 6.22i)17-s + ⋯
L(s)  = 1  + (−0.0739 − 0.703i)2-s + (−0.386 + 0.429i)3-s + (−0.489 + 0.103i)4-s + (0.162 − 0.0723i)5-s + (0.330 + 0.239i)6-s + (−0.535 + 0.844i)7-s + (0.109 + 0.336i)8-s + (−0.0348 − 0.331i)9-s + (−0.0628 − 0.108i)10-s + (−0.247 − 0.968i)11-s + (0.144 − 0.249i)12-s + (−0.722 + 0.525i)13-s + (0.633 + 0.313i)14-s + (−0.0317 + 0.0976i)15-s + (0.228 − 0.101i)16-s + (−0.158 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.346 - 0.938i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.346 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260703 + 0.374022i\)
\(L(\frac12)\) \(\approx\) \(0.260703 + 0.374022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
7 \( 1 + (1.41 - 2.23i)T \)
11 \( 1 + (0.820 + 3.21i)T \)
good5 \( 1 + (-0.363 + 0.161i)T + (3.34 - 3.71i)T^{2} \)
13 \( 1 + (2.60 - 1.89i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.654 - 6.22i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (4.92 + 1.04i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (3.00 - 5.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.61 + 4.95i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.289 + 0.128i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-2.18 - 2.42i)T + (-3.86 + 36.7i)T^{2} \)
41 \( 1 + (-2.86 - 8.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.62T + 43T^{2} \)
47 \( 1 + (7.69 + 1.63i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (-10.6 - 4.74i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (-7.74 + 1.64i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (3.35 - 1.49i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (7.15 + 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.67 + 3.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.02 - 1.70i)T + (66.6 - 29.6i)T^{2} \)
79 \( 1 + (-0.0695 - 0.662i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-7.09 - 5.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.97 + 5.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.7 + 9.23i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.40992984197977350862589785143, −10.40091028588899319829011028187, −9.685599463235096275145382878524, −8.866531493417312437939500680252, −7.999765410250553123341603714399, −6.31886011285844763213002673040, −5.70476823424427643335561997603, −4.43288712459619203853701298258, −3.34182798241863597063382309163, −2.01748695891146912723983979175, 0.28543454716528358712941897044, 2.42588746818709385311346460746, 4.20737543439755257169315210501, 5.12028233731250270864111652299, 6.32984101994090345081266914274, 7.08313469981839749948024241978, 7.69065784444916340756031943400, 8.887366683087826410671145343127, 10.12243176981458839084849407826, 10.35572917250743189739434907243

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