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Properties

Label 2-6030-201.200-c1-0-44
Degree $2$
Conductor $6030$
Sign $0.121 - 0.992i$
Analytic cond. $48.1497$
Root an. cond. $6.93900$
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 + 4-s + 5-s + 3.69i·7-s + 8-s + 10-s + 5.10·11-s + 6.05i·13-s + 3.69i·14-s + 16-s + 0.252i·17-s + 4.17·19-s + 20-s + 5.10·22-s − 5.65i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.39i·7-s + 0.353·8-s + 0.316·10-s + 1.53·11-s + 1.67i·13-s + 0.987i·14-s + 0.250·16-s + 0.0613i·17-s + 0.958·19-s + 0.223·20-s + 1.08·22-s − 1.17i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
Sign: $0.121 - 0.992i$
Analytic conductor: \(48.1497\)
Root analytic conductor: \(6.93900\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6030} (2411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6030,\ (\ :1/2),\ 0.121 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.913696089\)
\(L(\frac12)\) \(\approx\) \(3.913696089\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
67 \( 1 + (-6.06 - 5.50i)T \)
good7 \( 1 - 3.69iT - 7T^{2} \)
11 \( 1 - 5.10T + 11T^{2} \)
13 \( 1 - 6.05iT - 13T^{2} \)
17 \( 1 - 0.252iT - 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 - 9.12iT - 29T^{2} \)
31 \( 1 - 7.22iT - 31T^{2} \)
37 \( 1 + 5.83T + 37T^{2} \)
41 \( 1 + 9.99T + 41T^{2} \)
43 \( 1 - 5.80iT - 43T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 + 3.38T + 53T^{2} \)
59 \( 1 + 11.2iT - 59T^{2} \)
61 \( 1 + 2.77iT - 61T^{2} \)
71 \( 1 + 8.07iT - 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 + 8.22iT - 79T^{2} \)
83 \( 1 + 12.6iT - 83T^{2} \)
89 \( 1 - 2.69iT - 89T^{2} \)
97 \( 1 - 3.99iT - 97T^{2} \)
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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

−8.634366720700246567508335395509, −7.13657019235655530828016113551, −6.58485866330485018960350692716, −6.29320080051612579418314927032, −5.09974644448977863438507444362, −4.94398692397372693656176536484, −3.73562684884858951531251391561, −3.13448814056767649172365737759, −1.96521569315623276394850776645, −1.54996782822106979063967825805, 0.77352157494030169424186703797, 1.52354155169023185826703655570, 2.80649172456290090109153288779, 3.68377086082922754537573005537, 4.04241654194785772145129493494, 5.07327204808780911879145050482, 5.76270637627133625272336964298, 6.38392318955728755087478731366, 7.24573508441627629535885511443, 7.62431658280223559982534511960

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