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Properties

Label 2-462-11.4-c1-0-5
Degree $2$
Conductor $462$
Sign $0.970 + 0.241i$
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  + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.0359 − 0.0261i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 0.0444·10-s + (−0.154 − 3.31i)11-s − 12-s + (4.52 + 3.28i)13-s + (−0.309 + 0.951i)14-s + (0.0137 + 0.0422i)15-s + (−0.809 + 0.587i)16-s + (2.18 − 1.59i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.0160 − 0.0116i)5-s + (0.330 − 0.239i)6-s + (−0.116 − 0.359i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s − 0.0140·10-s + (−0.0464 − 0.998i)11-s − 0.288·12-s + (1.25 + 0.911i)13-s + (−0.0825 + 0.254i)14-s + (0.00354 + 0.0109i)15-s + (−0.202 + 0.146i)16-s + (0.531 − 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.970 + 0.241i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03588 - 0.126973i\)
\(L(\frac12)\) \(\approx\) \(1.03588 - 0.126973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.154 + 3.31i)T \)
good5 \( 1 + (-0.0359 + 0.0261i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-4.52 - 3.28i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.18 + 1.59i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.10 + 3.41i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 + (-1.87 - 5.77i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.18 - 5.21i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.777 + 2.39i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.13 + 6.57i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 + (-3.52 + 10.8i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.65 + 4.83i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.02 + 3.14i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.81 - 7.13i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.01T + 67T^{2} \)
71 \( 1 + (2.85 - 2.07i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.196 + 0.603i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.05 - 3.67i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.43 - 3.22i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 2.76T + 89T^{2} \)
97 \( 1 + (9.05 + 6.57i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.00720816722641482324476048695, −10.25701504361440727770108474899, −9.035480497553257161289518965545, −8.796387041628395708646439147102, −7.42044468392463847470044251459, −6.43706053207115640280997585524, −5.25491879941891449760270264478, −3.92560914762902276840878462547, −3.02767622492594222773017757000, −1.06913329991980446843529490238, 1.18435249656562747777614492179, 2.73253362468072086620813765661, 4.43617118096497190473137179731, 5.92345899619315522959463006870, 6.25680860346628464393421943076, 7.76078940595202639934426272260, 8.022171485198060976275196581348, 9.263976460923818380006549583536, 10.12647249814065233647282304883, 10.94137588495634236794875729001

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