Properties

Label 2-6030-201.200-c1-0-3
Degree $2$
Conductor $6030$
Sign $-0.968 + 0.249i$
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 + 1.26i·7-s − 8-s + 10-s + 1.70·11-s + 4.06i·13-s − 1.26i·14-s + 16-s + 3.79i·17-s − 5.81·19-s − 20-s − 1.70·22-s + 3.51i·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.478i·7-s − 0.353·8-s + 0.316·10-s + 0.515·11-s + 1.12i·13-s − 0.338i·14-s + 0.250·16-s + 0.920i·17-s − 1.33·19-s − 0.223·20-s − 0.364·22-s + 0.733i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
Sign: $-0.968 + 0.249i$
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: \((2,\ 6030,\ (\ :1/2),\ -0.968 + 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3867507429\)
\(L(\frac12)\) \(\approx\) \(0.3867507429\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.24 - 5.29i)T \)
good7 \( 1 - 1.26iT - 7T^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 - 4.06iT - 13T^{2} \)
17 \( 1 - 3.79iT - 17T^{2} \)
19 \( 1 + 5.81T + 19T^{2} \)
23 \( 1 - 3.51iT - 23T^{2} \)
29 \( 1 - 1.09iT - 29T^{2} \)
31 \( 1 - 9.21iT - 31T^{2} \)
37 \( 1 - 4.05T + 37T^{2} \)
41 \( 1 + 6.43T + 41T^{2} \)
43 \( 1 - 3.64iT - 43T^{2} \)
47 \( 1 + 9.46iT - 47T^{2} \)
53 \( 1 + 8.27T + 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 - 1.02iT - 61T^{2} \)
71 \( 1 + 6.95iT - 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 8.81iT - 79T^{2} \)
83 \( 1 + 6.03iT - 83T^{2} \)
89 \( 1 + 2.09iT - 89T^{2} \)
97 \( 1 - 7.10iT - 97T^{2} \)
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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

−8.553747520143418997695457435105, −7.951847556806353425498724019626, −6.89824342378063694682874942953, −6.63366738815134838269340543201, −5.79334799672850016666082264753, −4.80565225782074724940866221348, −3.99793506983885539912317288527, −3.24802766632274309359652206353, −2.07303816518832202366745754725, −1.44723753005406846472455274648, 0.14535474571013223668367360541, 0.976225785058278243332160147137, 2.27039158967064895621080859527, 3.04331013992066909343885882574, 4.04516938136893744643421312394, 4.64271005865202940018086173378, 5.76994933577319419465214212929, 6.38233988656142320779045928217, 7.18842417611395174576374481377, 7.73978994983777324976490157054

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