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Properties

Label 2-2970-9.4-c1-0-33
Degree $2$
Conductor $2970$
Sign $0.939 + 0.342i$
Analytic cond. $23.7155$
Root an. cond. $4.86986$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−0.5 − 0.866i)11-s + (2 − 3.46i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 2·19-s + (0.499 + 0.866i)20-s + (0.499 − 0.866i)22-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s + 3.99·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (0.188 + 0.327i)7-s − 0.353·8-s + 0.316·10-s + (−0.150 − 0.261i)11-s + (0.554 − 0.960i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.458·19-s + (0.111 + 0.193i)20-s + (0.106 − 0.184i)22-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s + 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(23.7155\)
Root analytic conductor: \(4.86986\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2970} (1981, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2970,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068102026\)
\(L(\frac12)\) \(\approx\) \(2.068102026\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.465876148507113971076851805881, −8.059162125610391295452188385592, −7.17964264317765695732297779561, −6.29160513198901125673414852987, −5.55745447033473197856676234209, −5.10458670779575195848069646675, −4.03335984711846781712001303800, −3.20627792663543249622237182019, −2.10276490852306445467047059351, −0.61648331548095152351104704150, 1.24683394166355372618329634764, 2.10527371663743790765946140555, 3.23091869555761032669870408399, 3.91097489900020388208664097765, 4.84741423610054210432916960034, 5.58414514137129143915930860149, 6.51228474815562703284985188201, 7.18506569991856063170080127367, 8.001237766350799376894650545120, 9.149905122007209956844544209935

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