Properties

Label 2-30e2-15.8-c3-0-1
Degree $2$
Conductor $900$
Sign $-0.986 + 0.161i$
Analytic cond. $53.1017$
Root an. cond. $7.28709$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−17.1 + 17.1i)7-s + 68.0i·11-s + (33.1 + 33.1i)13-s + (−15.5 − 15.5i)17-s + 40.1i·19-s + (52.1 − 52.1i)23-s + 97.9·29-s − 206.·31-s + (238. − 238. i)37-s − 43.1i·41-s + (−255. − 255. i)43-s + (−134. − 134. i)47-s − 248. i·49-s + (−458. + 458. i)53-s − 748.·59-s + ⋯
L(s)  = 1  + (−0.928 + 0.928i)7-s + 1.86i·11-s + (0.707 + 0.707i)13-s + (−0.222 − 0.222i)17-s + 0.484i·19-s + (0.473 − 0.473i)23-s + 0.627·29-s − 1.19·31-s + (1.05 − 1.05i)37-s − 0.164i·41-s + (−0.905 − 0.905i)43-s + (−0.418 − 0.418i)47-s − 0.724i·49-s + (−1.18 + 1.18i)53-s − 1.65·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.986 + 0.161i$
Analytic conductor: \(53.1017\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :3/2),\ -0.986 + 0.161i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7252689384\)
\(L(\frac12)\) \(\approx\) \(0.7252689384\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (17.1 - 17.1i)T - 343iT^{2} \)
11 \( 1 - 68.0iT - 1.33e3T^{2} \)
13 \( 1 + (-33.1 - 33.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (15.5 + 15.5i)T + 4.91e3iT^{2} \)
19 \( 1 - 40.1iT - 6.85e3T^{2} \)
23 \( 1 + (-52.1 + 52.1i)T - 1.21e4iT^{2} \)
29 \( 1 - 97.9T + 2.43e4T^{2} \)
31 \( 1 + 206.T + 2.97e4T^{2} \)
37 \( 1 + (-238. + 238. i)T - 5.06e4iT^{2} \)
41 \( 1 + 43.1iT - 6.89e4T^{2} \)
43 \( 1 + (255. + 255. i)T + 7.95e4iT^{2} \)
47 \( 1 + (134. + 134. i)T + 1.03e5iT^{2} \)
53 \( 1 + (458. - 458. i)T - 1.48e5iT^{2} \)
59 \( 1 + 748.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + (133. - 133. i)T - 3.00e5iT^{2} \)
71 \( 1 - 899. iT - 3.57e5T^{2} \)
73 \( 1 + (26.5 + 26.5i)T + 3.89e5iT^{2} \)
79 \( 1 + 112. iT - 4.93e5T^{2} \)
83 \( 1 + (644. - 644. i)T - 5.71e5iT^{2} \)
89 \( 1 - 554.T + 7.04e5T^{2} \)
97 \( 1 + (-1.24e3 + 1.24e3i)T - 9.12e5iT^{2} \)
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

−9.968786781233269479707333936115, −9.339640786661826042343335417998, −8.726564689760703294054116973020, −7.48733491809741411799276656207, −6.73091833047101448598294376883, −5.94585742060679880344260336712, −4.85238395417734512526168988865, −3.90242906659273310501323475729, −2.65444924713947472197257396130, −1.70032246664575322560969828214, 0.20029059867829206105051361313, 1.15685524573008074933237484902, 3.15828481397898227107454719676, 3.45251098295780027560873215492, 4.82315684866160162344080637909, 6.05643595494048267243391826560, 6.47930086958755028800474743740, 7.66762059864145797301416535889, 8.428358172347610281232858781101, 9.284945161617975574299712941166

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