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Properties

Label 2-2970-99.65-c1-0-37
Degree $2$
Conductor $2970$
Sign $0.996 + 0.0847i$
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 + (3.18 + 1.83i)7-s + 0.999·8-s + 0.999i·10-s + (2.35 − 2.34i)11-s + (−1.88 + 1.09i)13-s + (−3.18 + 1.83i)14-s + (−0.5 + 0.866i)16-s + 1.68·17-s − 7.86i·19-s + (−0.866 − 0.499i)20-s + (0.851 + 3.20i)22-s + (4.34 − 2.50i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.387 − 0.223i)5-s + (1.20 + 0.694i)7-s + 0.353·8-s + 0.316i·10-s + (0.708 − 0.705i)11-s + (−0.524 + 0.302i)13-s + (−0.851 + 0.491i)14-s + (−0.125 + 0.216i)16-s + 0.408·17-s − 1.80i·19-s + (−0.193 − 0.111i)20-s + (0.181 + 0.683i)22-s + (0.905 − 0.522i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $0.996 + 0.0847i$
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: \((2,\ 2970,\ (\ :1/2),\ 0.996 + 0.0847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.927448377\)
\(L(\frac12)\) \(\approx\) \(1.927448377\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.35 + 2.34i)T \)
good7 \( 1 + (-3.18 - 1.83i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (1.88 - 1.09i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.68T + 17T^{2} \)
19 \( 1 + 7.86iT - 19T^{2} \)
23 \( 1 + (-4.34 + 2.50i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.65 + 6.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.78 - 4.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.84T + 37T^{2} \)
41 \( 1 + (2.91 + 5.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.89 - 1.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.46 + 2.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.66iT - 53T^{2} \)
59 \( 1 + (-12.1 + 6.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.93 + 4.57i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.87 - 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.93iT - 71T^{2} \)
73 \( 1 - 4.80iT - 73T^{2} \)
79 \( 1 + (12.4 + 7.16i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.79 - 4.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 + (7.38 - 12.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.509748313685952159357154458074, −8.365437679908905635156141821255, −7.02991286906449793694517113768, −6.69573185898411308032418682196, −5.53998028948178335134264575402, −5.07662962581277053875191950049, −4.34984511416079793780781135334, −2.90815778965250082271852099859, −1.89781763595628667648061677586, −0.77238325750220096535108057808, 1.26754657742110019205044555767, 1.75922144115336477756408703945, 3.04733853571092123159996009211, 3.95472697913911701152504534179, 4.77000769951879311002939784649, 5.51300769472512205537001974444, 6.68202863882625786767698911999, 7.44279559341187818062177300910, 7.987470202224423863703575570457, 8.793629029311416075546347849111

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