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Properties

Label 2-2970-99.65-c1-0-23
Degree $2$
Conductor $2970$
Sign $0.709 - 0.705i$
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 + (0.866 − 0.5i)5-s + (0.915 + 0.528i)7-s + 0.999·8-s + 0.999i·10-s + (1.83 + 2.75i)11-s + (4.33 − 2.50i)13-s + (−0.915 + 0.528i)14-s + (−0.5 + 0.866i)16-s − 4.08·17-s − 1.38i·19-s + (−0.866 − 0.499i)20-s + (−3.30 + 0.212i)22-s + (4.52 − 2.61i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (0.346 + 0.199i)7-s + 0.353·8-s + 0.316i·10-s + (0.554 + 0.832i)11-s + (1.20 − 0.694i)13-s + (−0.244 + 0.141i)14-s + (−0.125 + 0.216i)16-s − 0.989·17-s − 0.318i·19-s + (−0.193 − 0.111i)20-s + (−0.705 + 0.0453i)22-s + (0.943 − 0.544i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $0.709 - 0.705i$
Analytic conductor: \(23.7155\)
Root analytic conductor: \(4.86986\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2970} (791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2970,\ (\ :1/2),\ 0.709 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.844935519\)
\(L(\frac12)\) \(\approx\) \(1.844935519\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-1.83 - 2.75i)T \)
good7 \( 1 + (-0.915 - 0.528i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (-4.33 + 2.50i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.08T + 17T^{2} \)
19 \( 1 + 1.38iT - 19T^{2} \)
23 \( 1 + (-4.52 + 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.32 - 5.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.35 - 5.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.98T + 37T^{2} \)
41 \( 1 + (-1.10 - 1.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.85 - 1.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.01 + 0.587i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.0346iT - 53T^{2} \)
59 \( 1 + (-9.41 + 5.43i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.91 + 2.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.21 + 7.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.03iT - 71T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 + (8.56 + 4.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.65 - 2.86i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.71iT - 89T^{2} \)
97 \( 1 + (2.76 - 4.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.729862626957995943067540886587, −8.289643425591102745784075253467, −7.21120897769360918405678868816, −6.64275063074618206670861756428, −5.91129425314863557397530795998, −5.00717563815160849228421291174, −4.42296023060268212706851668704, −3.21151989395039200752068049772, −1.95468796269537763490482864441, −0.976215024332197220624951184605, 0.884892521026240888822812930319, 1.84471959210591184032526670642, 2.86106659777003751910025238663, 3.89714347681364135398333704000, 4.42251098464407225374210218970, 5.77832118622164773452640761295, 6.28454268939657560101825580575, 7.22796807482471191790129444872, 8.073139229835043826420999064384, 8.876695813001736978775715204989

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