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Properties

Label 2-6030-201.200-c1-0-63
Degree $2$
Conductor $6030$
Sign $0.845 + 0.534i$
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 − 2.64i·7-s + 8-s + 10-s + 1.34·11-s − 2.35i·13-s − 2.64i·14-s + 16-s + 6.24i·17-s + 3.10·19-s + 20-s + 1.34·22-s − 7.74i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.997i·7-s + 0.353·8-s + 0.316·10-s + 0.405·11-s − 0.653i·13-s − 0.705i·14-s + 0.250·16-s + 1.51i·17-s + 0.712·19-s + 0.223·20-s + 0.286·22-s − 1.61i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
Sign: $0.845 + 0.534i$
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: \((2,\ 6030,\ (\ :1/2),\ 0.845 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.850306151\)
\(L(\frac12)\) \(\approx\) \(3.850306151\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.424 + 8.17i)T \)
good7 \( 1 + 2.64iT - 7T^{2} \)
11 \( 1 - 1.34T + 11T^{2} \)
13 \( 1 + 2.35iT - 13T^{2} \)
17 \( 1 - 6.24iT - 17T^{2} \)
19 \( 1 - 3.10T + 19T^{2} \)
23 \( 1 + 7.74iT - 23T^{2} \)
29 \( 1 - 8.46iT - 29T^{2} \)
31 \( 1 - 7.92iT - 31T^{2} \)
37 \( 1 - 9.02T + 37T^{2} \)
41 \( 1 - 7.85T + 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 - 3.95iT - 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 - 5.88iT - 59T^{2} \)
61 \( 1 + 3.93iT - 61T^{2} \)
71 \( 1 + 8.97iT - 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 2.89iT - 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 - 9.61iT - 89T^{2} \)
97 \( 1 - 1.56iT - 97T^{2} \)
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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

−7.88560892802574565645453535301, −7.17018487917303571194191934387, −6.47306382845996738786395315176, −5.94989682385423802196955706882, −5.04132810230054320373615066860, −4.39283117233909780410702917928, −3.60224669251609343897724425754, −2.92380679998149970724347310895, −1.76554821917648750536937897330, −0.886926308736095970468623581135, 1.07140161654974848576490038949, 2.27161840497193194371947901160, 2.71915246002479270122897337250, 3.77327465983968996461519840773, 4.58938980300682292378308216884, 5.32028709607367338871017645759, 5.97653339368153697701443535215, 6.42720409556369562873666639653, 7.50717306646061561431508685469, 7.83842911159575960867765878623

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