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Properties

Label 2-462-77.76-c1-0-1
Degree $2$
Conductor $462$
Sign $-0.752 - 0.658i$
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  − i·2-s + i·3-s − 4-s + 3.06i·5-s + 6-s + (−2.37 − 1.16i)7-s + i·8-s − 9-s + 3.06·10-s + (−3.20 − 0.857i)11-s − i·12-s − 0.338·13-s + (−1.16 + 2.37i)14-s − 3.06·15-s + 16-s + 0.314·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 1.36i·5-s + 0.408·6-s + (−0.897 − 0.441i)7-s + 0.353i·8-s − 0.333·9-s + 0.968·10-s + (−0.966 − 0.258i)11-s − 0.288i·12-s − 0.0939·13-s + (−0.312 + 0.634i)14-s − 0.790·15-s + 0.250·16-s + 0.0762·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152970 + 0.406863i\)
\(L(\frac12)\) \(\approx\) \(0.152970 + 0.406863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
7 \( 1 + (2.37 + 1.16i)T \)
11 \( 1 + (3.20 + 0.857i)T \)
good5 \( 1 - 3.06iT - 5T^{2} \)
13 \( 1 + 0.338T + 13T^{2} \)
17 \( 1 - 0.314T + 17T^{2} \)
19 \( 1 + 6.09T + 19T^{2} \)
23 \( 1 + 3.37T + 23T^{2} \)
29 \( 1 - 4.40iT - 29T^{2} \)
31 \( 1 - 0.722iT - 31T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 - 8.09T + 41T^{2} \)
43 \( 1 - 4.12iT - 43T^{2} \)
47 \( 1 - 8.77iT - 47T^{2} \)
53 \( 1 - 13.5T + 53T^{2} \)
59 \( 1 - 4.67iT - 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 1.68T + 73T^{2} \)
79 \( 1 + 6.86iT - 79T^{2} \)
83 \( 1 + 7.11T + 83T^{2} \)
89 \( 1 + 2.28iT - 89T^{2} \)
97 \( 1 + 5.08iT - 97T^{2} \)
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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

−10.84860211778570684196782252700, −10.67845000581055212789720609992, −9.989388581767259248270494140673, −8.977769770294834484622807244399, −7.78556304293188795233290604421, −6.71852920035300346412599521363, −5.76298347752744110578319363023, −4.30249531982013937154616146367, −3.30045001293028192007591609758, −2.51233501805385342400685252479, 0.24992593030478936154120329787, 2.25473283070807984301863501884, 4.04183238923858384016687351719, 5.21897081459666285617372182810, 5.95375288595390618643634762893, 6.98917547239946865176401550316, 8.099859275851530204072238485213, 8.664385138763281765861101289166, 9.540580376201025781790600090934, 10.51091418858700008136915607920

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