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Properties

Label 2-462-21.20-c1-0-6
Degree $2$
Conductor $462$
Sign $0.487 - 0.872i$
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 + (−1.61 − 0.618i)3-s − 4-s − 3.23·5-s + (0.618 − 1.61i)6-s + (0.381 − 2.61i)7-s − i·8-s + (2.23 + 2.00i)9-s − 3.23i·10-s − i·11-s + (1.61 + 0.618i)12-s + 6i·13-s + (2.61 + 0.381i)14-s + (5.23 + 2.00i)15-s + 16-s + 7.23·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.934 − 0.356i)3-s − 0.5·4-s − 1.44·5-s + (0.252 − 0.660i)6-s + (0.144 − 0.989i)7-s − 0.353i·8-s + (0.745 + 0.666i)9-s − 1.02i·10-s − 0.301i·11-s + (0.467 + 0.178i)12-s + 1.66i·13-s + (0.699 + 0.102i)14-s + (1.35 + 0.516i)15-s + 0.250·16-s + 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.487 - 0.872i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607834 + 0.356572i\)
\(L(\frac12)\) \(\approx\) \(0.607834 + 0.356572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.618i)T \)
7 \( 1 + (-0.381 + 2.61i)T \)
11 \( 1 + iT \)
good5 \( 1 + 3.23T + 5T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 7.23T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 1.23iT - 23T^{2} \)
29 \( 1 - 0.472iT - 29T^{2} \)
31 \( 1 - 0.763iT - 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 1.52T + 47T^{2} \)
53 \( 1 - 7.70iT - 53T^{2} \)
59 \( 1 - 3.23T + 59T^{2} \)
61 \( 1 + 3.52iT - 61T^{2} \)
67 \( 1 - 1.52T + 67T^{2} \)
71 \( 1 + 6.18iT - 71T^{2} \)
73 \( 1 - 14.1iT - 73T^{2} \)
79 \( 1 - 7.23T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 2.47iT - 97T^{2} \)
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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

−11.29962584822585828644673051019, −10.51113343226774783540031280369, −9.350064208960959844459114412993, −7.960195100297008361430514007303, −7.50927110551669740277263008710, −6.77201429867421611585224717694, −5.63996305178035765135593991040, −4.41055849278307176460139108080, −3.80645174564738535204300583265, −1.01607175494390905174026577253, 0.69097073518968889309072920179, 2.96460290260980045292561963437, 3.97153044460235755914528851188, 5.11231936387832528042773213988, 5.82434976170426261929661264938, 7.45913893790855839235613836933, 8.141890011241811922362863355993, 9.311694145412143987230738137314, 10.28957020180055898715842435896, 11.02084451659308592347498418975

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