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Properties

Label 2-462-21.20-c1-0-4
Degree $2$
Conductor $462$
Sign $-0.966 - 0.255i$
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 + (1.51 + 0.848i)3-s − 4-s − 1.69·5-s + (−0.848 + 1.51i)6-s + (−2.56 + 0.662i)7-s − i·8-s + (1.56 + 2.56i)9-s − 1.69i·10-s + i·11-s + (−1.51 − 0.848i)12-s + 4.34i·13-s + (−0.662 − 2.56i)14-s + (−2.56 − 1.43i)15-s + 16-s − 4.71·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.871 + 0.489i)3-s − 0.5·4-s − 0.758·5-s + (−0.346 + 0.616i)6-s + (−0.968 + 0.250i)7-s − 0.353i·8-s + (0.520 + 0.853i)9-s − 0.536i·10-s + 0.301i·11-s + (−0.435 − 0.244i)12-s + 1.20i·13-s + (−0.176 − 0.684i)14-s + (−0.661 − 0.371i)15-s + 0.250·16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.966 - 0.255i$
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: \((2,\ 462,\ (\ :1/2),\ -0.966 - 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141364 + 1.08674i\)
\(L(\frac12)\) \(\approx\) \(0.141364 + 1.08674i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 - 0.848i)T \)
7 \( 1 + (2.56 - 0.662i)T \)
11 \( 1 - iT \)
good5 \( 1 + 1.69T + 5T^{2} \)
13 \( 1 - 4.34iT - 13T^{2} \)
17 \( 1 + 4.71T + 17T^{2} \)
19 \( 1 - 6.41iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 1.32iT - 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 - 7.36T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + 7.36T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 1.69iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 9.36iT - 71T^{2} \)
73 \( 1 - 8.68iT - 73T^{2} \)
79 \( 1 + 6.87T + 79T^{2} \)
83 \( 1 - 6.41T + 83T^{2} \)
89 \( 1 - 6.78T + 89T^{2} \)
97 \( 1 + 6.04iT - 97T^{2} \)
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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

−11.44467406827598466098783119162, −10.24566051050024296827313048184, −9.449942387230575302123370943964, −8.742617335930204245237438822726, −7.88758621511852205114064560216, −6.96413624389124117071729435857, −6.01950419781990949957967760309, −4.37695202404596451734527674660, −3.94924233610711625766949691895, −2.44312017933496780989630380696, 0.60386274907848911777396302321, 2.58029276408909777813757558934, 3.39144784555083862145230364757, 4.36334750987217848319265546399, 6.01901344488902935773333721111, 7.23082667264515860124191540009, 7.922847683308803693029084583366, 9.024759771380888967785764765321, 9.562230278574602179381088689915, 10.74756086549290282068530294838

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