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Properties

Label 2-462-33.32-c1-0-0
Degree $2$
Conductor $462$
Sign $-0.933 - 0.359i$
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  − 2-s + (−0.742 + 1.56i)3-s + 4-s − 3.23i·5-s + (0.742 − 1.56i)6-s + i·7-s − 8-s + (−1.89 − 2.32i)9-s + 3.23i·10-s + (−3.30 + 0.248i)11-s + (−0.742 + 1.56i)12-s + 5.12i·13-s − i·14-s + (5.05 + 2.39i)15-s + 16-s − 2.90·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.428 + 0.903i)3-s + 0.5·4-s − 1.44i·5-s + (0.303 − 0.638i)6-s + 0.377i·7-s − 0.353·8-s + (−0.632 − 0.774i)9-s + 1.02i·10-s + (−0.997 + 0.0749i)11-s + (−0.214 + 0.451i)12-s + 1.42i·13-s − 0.267i·14-s + (1.30 + 0.619i)15-s + 0.250·16-s − 0.703·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.933 - 0.359i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0480894 + 0.258397i\)
\(L(\frac12)\) \(\approx\) \(0.0480894 + 0.258397i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.742 - 1.56i)T \)
7 \( 1 - iT \)
11 \( 1 + (3.30 - 0.248i)T \)
good5 \( 1 + 3.23iT - 5T^{2} \)
13 \( 1 - 5.12iT - 13T^{2} \)
17 \( 1 + 2.90T + 17T^{2} \)
19 \( 1 - 0.590iT - 19T^{2} \)
23 \( 1 - 9.26iT - 23T^{2} \)
29 \( 1 + 0.996T + 29T^{2} \)
31 \( 1 + 4.39T + 31T^{2} \)
37 \( 1 + 8.10T + 37T^{2} \)
41 \( 1 + 9.80T + 41T^{2} \)
43 \( 1 - 1.94iT - 43T^{2} \)
47 \( 1 + 7.19iT - 47T^{2} \)
53 \( 1 - 0.204iT - 53T^{2} \)
59 \( 1 - 6.09iT - 59T^{2} \)
61 \( 1 + 5.31iT - 61T^{2} \)
67 \( 1 - 8.79T + 67T^{2} \)
71 \( 1 - 6.62iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 - 5.49iT - 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 + 1.28iT - 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−11.47201500257950633064821394565, −10.39944512983248434662607921191, −9.416559484626104893562317541037, −9.022990207682891436064648430068, −8.192377382077542465260177321300, −6.87429680105909430060844240312, −5.52460211594342064786612302619, −4.95553402661444940145841411143, −3.71451044902018319497690619263, −1.81041181629623266742299172254, 0.20095121924102129434636099367, 2.26698922419371353744628073815, 3.14666330810982099816707864955, 5.22352409357862354153903638182, 6.36746688855401366704972327730, 7.00500619874898397484070360190, 7.79432541514861695878338608226, 8.550708989451714216510811456567, 10.23790820341713571304472035994, 10.63782152889397875653984652748

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