↑

Properties

Label 2-2970-99.32-c1-0-39
Degree $2$
Conductor $2970$
Sign $-0.985 - 0.168i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (−0.0653 + 0.0377i)7-s + 0.999·8-s + 0.999i·10-s + (1.32 + 3.04i)11-s + (1.75 + 1.01i)13-s + (0.0653 + 0.0377i)14-s + (−0.5 − 0.866i)16-s − 5.56·17-s − 1.11i·19-s + (0.866 − 0.499i)20-s + (1.97 − 2.66i)22-s + (−0.679 − 0.392i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.0247 + 0.0142i)7-s + 0.353·8-s + 0.316i·10-s + (0.399 + 0.916i)11-s + (0.485 + 0.280i)13-s + (0.0174 + 0.0100i)14-s + (−0.125 − 0.216i)16-s − 1.35·17-s − 0.255i·19-s + (0.193 − 0.111i)20-s + (0.420 − 0.568i)22-s + (−0.141 − 0.0818i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $-0.985 - 0.168i$
Analytic conductor: \(23.7155\)
Root analytic conductor: \(4.86986\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2970} (2771, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2970,\ (\ :1/2),\ -0.985 - 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2252836517\)
\(L(\frac12)\) \(\approx\) \(0.2252836517\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 3.04i)T \)
good7 \( 1 + (0.0653 - 0.0377i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-1.75 - 1.01i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.56T + 17T^{2} \)
19 \( 1 + 1.11iT - 19T^{2} \)
23 \( 1 + (0.679 + 0.392i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.10 + 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.23 - 7.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 + (-1.63 + 2.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.6 + 6.14i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.34 - 3.66i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.64iT - 53T^{2} \)
59 \( 1 + (10.9 + 6.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.86 + 4.53i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.11 - 1.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 6.18iT - 73T^{2} \)
79 \( 1 + (8.37 - 4.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.11 + 1.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.50iT - 89T^{2} \)
97 \( 1 + (8.14 + 14.1i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.524437122107437249096700204969, −7.67885557479619915043927676999, −6.96203104288931522334979227265, −6.19814630766189153135793435838, −4.96549723148888001466649557921, −4.29227023304400608318270576485, −3.58669223761820668071743133154, −2.37679775127519911735440217056, −1.55460879061773526754346077233, −0.084508610684680733267414135291, 1.26834358854074165783759613285, 2.63976046512543536176521488252, 3.75072038243506071239258655064, 4.41213437391971850312667468779, 5.59920256036436677164918637680, 6.13335107450155310178920352360, 6.92041290001941208996260954728, 7.65491026347179059197243194234, 8.385609379401896139778331085417, 8.987343107178290494307856143498

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