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Properties

Label 2-6030-201.200-c1-0-54
Degree $2$
Conductor $6030$
Sign $0.897 + 0.440i$
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 + 5.18i·7-s + 8-s − 10-s − 5.96·11-s − 5.86i·13-s + 5.18i·14-s + 16-s − 4.42i·17-s + 5.41·19-s − 20-s − 5.96·22-s − 2.69i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.95i·7-s + 0.353·8-s − 0.316·10-s − 1.79·11-s − 1.62i·13-s + 1.38i·14-s + 0.250·16-s − 1.07i·17-s + 1.24·19-s − 0.223·20-s − 1.27·22-s − 0.562i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
Sign: $0.897 + 0.440i$
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: \((2,\ 6030,\ (\ :1/2),\ 0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.280493053\)
\(L(\frac12)\) \(\approx\) \(2.280493053\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.18 + 3.91i)T \)
good7 \( 1 - 5.18iT - 7T^{2} \)
11 \( 1 + 5.96T + 11T^{2} \)
13 \( 1 + 5.86iT - 13T^{2} \)
17 \( 1 + 4.42iT - 17T^{2} \)
19 \( 1 - 5.41T + 19T^{2} \)
23 \( 1 + 2.69iT - 23T^{2} \)
29 \( 1 + 6.15iT - 29T^{2} \)
31 \( 1 - 3.36iT - 31T^{2} \)
37 \( 1 - 4.86T + 37T^{2} \)
41 \( 1 - 4.68T + 41T^{2} \)
43 \( 1 - 12.5iT - 43T^{2} \)
47 \( 1 + 1.66iT - 47T^{2} \)
53 \( 1 - 0.782T + 53T^{2} \)
59 \( 1 + 5.16iT - 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
71 \( 1 - 5.99iT - 71T^{2} \)
73 \( 1 - 4.25T + 73T^{2} \)
79 \( 1 - 1.83iT - 79T^{2} \)
83 \( 1 + 9.73iT - 83T^{2} \)
89 \( 1 - 15.2iT - 89T^{2} \)
97 \( 1 + 6.02iT - 97T^{2} \)
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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

−7.988762171600070466547652937154, −7.51535709837348073531968115246, −6.33083170949191001420957548095, −5.65505665489589298125694568811, −5.17647144684423637405971063077, −4.75963477271980128707185971996, −3.18917044977633453384533456162, −2.87106302039472300674140277209, −2.28739080528606285640982913150, −0.53957125186804389033477568960, 0.905843417216140484322585595997, 1.97214590049130125989603107398, 3.12294780132386922690648221857, 3.87758934368799026434922770608, 4.34385024536181618167573074436, 5.11029980326614985218134584238, 5.91185784527655211108104585272, 6.94128330676047171589904716075, 7.35043055431757162566621971891, 7.75247568430610395318885023584

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