↑

Properties

Label 2-252-252.95-c1-0-21
Degree $2$
Conductor $252$
Sign $0.775 + 0.631i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + (−0.250 − 1.39i)2-s + (1.71 + 0.228i)3-s + (−1.87 + 0.696i)4-s + 1.07i·5-s + (−0.111 − 2.44i)6-s + (2.47 + 0.938i)7-s + (1.43 + 2.43i)8-s + (2.89 + 0.783i)9-s + (1.50 − 0.269i)10-s − 1.25·11-s + (−3.37 + 0.767i)12-s + (2.30 − 3.99i)13-s + (0.688 − 3.67i)14-s + (−0.246 + 1.85i)15-s + (3.03 − 2.61i)16-s + (−0.666 − 0.384i)17-s + ⋯
L(s)  = 1  + (−0.176 − 0.984i)2-s + (0.991 + 0.131i)3-s + (−0.937 + 0.348i)4-s + 0.482i·5-s + (−0.0456 − 0.998i)6-s + (0.934 + 0.354i)7-s + (0.508 + 0.861i)8-s + (0.965 + 0.261i)9-s + (0.474 − 0.0853i)10-s − 0.377·11-s + (−0.975 + 0.221i)12-s + (0.639 − 1.10i)13-s + (0.183 − 0.982i)14-s + (−0.0635 + 0.478i)15-s + (0.757 − 0.652i)16-s + (−0.161 − 0.0933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45394 - 0.517508i\)
\(L(\frac12)\) \(\approx\) \(1.45394 - 0.517508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.250 + 1.39i)T \)
3 \( 1 + (-1.71 - 0.228i)T \)
7 \( 1 + (-2.47 - 0.938i)T \)
good5 \( 1 - 1.07iT - 5T^{2} \)
11 \( 1 + 1.25T + 11T^{2} \)
13 \( 1 + (-2.30 + 3.99i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.666 + 0.384i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.65 - 0.952i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.15T + 23T^{2} \)
29 \( 1 + (-2.72 + 1.57i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.15 - 4.70i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.910 + 1.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.58 + 4.37i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.74 + 3.89i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.969 + 1.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.05 + 2.92i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.01 - 6.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.64 - 9.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.65 - 2.68i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.05T + 71T^{2} \)
73 \( 1 + (-4.18 + 7.25i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.4 + 6.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.69 - 2.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.43 + 0.827i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.231 - 0.401i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.94149322203445550333429314712, −10.60322989715543032735073200287, −10.40246787236666008950456810218, −8.953842483211439780494098420410, −8.324072770174502085017584819908, −7.47618151317143949217975461886, −5.50063420234648551378791284205, −4.17228200548251740213929524297, −3.02977158447176402277262948729, −1.86578723705574534473650535405, 1.67060793143685548427297553246, 3.96749498139760524178256444892, 4.79100350163605451141020466838, 6.31330127900792664813301877544, 7.43474272635541037379225918631, 8.256596201696668578765379712265, 8.882012109437473941326045849729, 9.857787016120988913251273969100, 11.03779436006680280537339731421, 12.49556581097473729934457087265

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