Properties

Label 2-2940-5.4-c1-0-24
Degree $2$
Conductor $2940$
Sign $0.894 + 0.447i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
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  + i·3-s + (−1 + 2i)5-s − 9-s − 4·11-s + (−2 − i)15-s − 4i·17-s − 4i·23-s + (−3 − 4i)25-s − i·27-s + 6·29-s − 4·31-s − 4i·33-s − 8i·37-s + 10·41-s − 4i·43-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.447 + 0.894i)5-s − 0.333·9-s − 1.20·11-s + (−0.516 − 0.258i)15-s − 0.970i·17-s − 0.834i·23-s + (−0.600 − 0.800i)25-s − 0.192i·27-s + 1.11·29-s − 0.718·31-s − 0.696i·33-s − 1.31i·37-s + 1.56·41-s − 0.609i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.017452045\)
\(L(\frac12)\) \(\approx\) \(1.017452045\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 + (1 - 2i)T \)
7 \( 1 \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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

−8.738659136578894848488510340048, −7.77839136969694624616842462399, −7.37719524432752333665482498017, −6.41051605657725976624063571491, −5.59213847209174319582998412860, −4.73641686312773950260903337717, −3.96588714301193085480673723765, −2.91083144254592597922393623083, −2.43433062179563664002280813199, −0.38912957418605204233968045201, 0.949879953779989810366524534503, 2.02860675558636914170257481991, 3.13662821620282539175315932176, 4.12099934700969364825161472499, 5.05627106343126384645776604662, 5.64312235629762590557318185782, 6.57169978731651954784374164866, 7.49930856919394887975895744223, 8.111349951585440869201714362962, 8.515115661106472199158030756414

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