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Properties

Label 2-2970-99.65-c1-0-34
Degree $2$
Conductor $2970$
Sign $0.952 - 0.305i$
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 + (3.61 + 2.08i)7-s + 0.999·8-s − 0.999i·10-s + (1.11 − 3.12i)11-s + (3.39 − 1.96i)13-s + (−3.61 + 2.08i)14-s + (−0.5 + 0.866i)16-s + 1.14·17-s + 4.10i·19-s + (0.866 + 0.499i)20-s + (2.14 + 2.52i)22-s + (5.50 − 3.17i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.387 + 0.223i)5-s + (1.36 + 0.789i)7-s + 0.353·8-s − 0.316i·10-s + (0.337 − 0.941i)11-s + (0.942 − 0.544i)13-s + (−0.966 + 0.558i)14-s + (−0.125 + 0.216i)16-s + 0.277·17-s + 0.942i·19-s + (0.193 + 0.111i)20-s + (0.457 + 0.539i)22-s + (1.14 − 0.662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $0.952 - 0.305i$
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: \((2,\ 2970,\ (\ :1/2),\ 0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.795617009\)
\(L(\frac12)\) \(\approx\) \(1.795617009\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 3.12i)T \)
good7 \( 1 + (-3.61 - 2.08i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (-3.39 + 1.96i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 - 4.10iT - 19T^{2} \)
23 \( 1 + (-5.50 + 3.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.62 + 2.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.55 + 7.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.54T + 37T^{2} \)
41 \( 1 + (3.46 + 5.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.59 + 4.96i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.36 - 4.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.65iT - 53T^{2} \)
59 \( 1 + (-2.56 + 1.48i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.80 + 3.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.70 + 8.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.97iT - 71T^{2} \)
73 \( 1 + 15.4iT - 73T^{2} \)
79 \( 1 + (3.43 + 1.98i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.80 + 4.86i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.0iT - 89T^{2} \)
97 \( 1 + (1.13 - 1.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.790111128491494478980480503467, −7.893641004071869311088449566168, −7.67110864684643638032669916940, −6.34902567926536888249662533870, −5.84677735002645144041468697301, −5.11899936391497479452194225102, −4.15595493437646984254429261898, −3.20363027049773381619174317654, −1.92771818180129685987650457560, −0.797109866231215818295790344453, 1.15032741855729615921722636409, 1.62621930128023839084783626851, 3.04695576858877668645306717060, 4.02667147564409359481404942649, 4.64920562617979327683890376109, 5.29036785242025283491590686545, 6.89050793184085225606539225708, 7.16970908847718249217817769476, 8.145803976081382028515913350282, 8.680836397385729008376010326845

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