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Properties

Label 2-6030-201.200-c1-0-62
Degree $2$
Conductor $6030$
Sign $-0.271 + 0.962i$
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 + 2.81i·7-s + 8-s − 10-s − 5.33·11-s + 0.767i·13-s + 2.81i·14-s + 16-s + 7.32i·17-s − 5.88·19-s − 20-s − 5.33·22-s − 8.27i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.06i·7-s + 0.353·8-s − 0.316·10-s − 1.60·11-s + 0.212i·13-s + 0.752i·14-s + 0.250·16-s + 1.77i·17-s − 1.34·19-s − 0.223·20-s − 1.13·22-s − 1.72i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
Sign: $-0.271 + 0.962i$
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: \((2,\ 6030,\ (\ :1/2),\ -0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7417094936\)
\(L(\frac12)\) \(\approx\) \(0.7417094936\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.71 - 2.73i)T \)
good7 \( 1 - 2.81iT - 7T^{2} \)
11 \( 1 + 5.33T + 11T^{2} \)
13 \( 1 - 0.767iT - 13T^{2} \)
17 \( 1 - 7.32iT - 17T^{2} \)
19 \( 1 + 5.88T + 19T^{2} \)
23 \( 1 + 8.27iT - 23T^{2} \)
29 \( 1 + 1.09iT - 29T^{2} \)
31 \( 1 - 4.54iT - 31T^{2} \)
37 \( 1 + 9.47T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + 11.9iT - 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 + 13.3iT - 61T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 - 8.36T + 73T^{2} \)
79 \( 1 - 9.70iT - 79T^{2} \)
83 \( 1 + 2.07iT - 83T^{2} \)
89 \( 1 + 4.36iT - 89T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−8.056963007984250624143616776010, −6.95814322448599146600702644462, −6.38668972927140109964804347646, −5.58838933685608407644795118629, −5.07864327013569575226375682897, −4.19827666053207891010977454076, −3.51074890174535709818974099371, −2.35123711995595789376354645415, −2.11539187446441541619656759542, −0.14851205362908860547421783310, 1.05924346065079632199741611877, 2.48559130069000684258808877730, 3.02960841787655921941419477473, 4.04511557899174201358798404755, 4.57115502478878183219390531201, 5.34964359940326118695417385444, 5.98679190106516265462445004950, 7.10078013843102363911906575102, 7.49218201371296690559308353191, 7.87707425082425592461857808666

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