↑

Properties

Label 2-252-28.19-c1-0-1
Degree $2$
Conductor $252$
Sign $-0.949 - 0.312i$
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  + (−0.850 + 1.12i)2-s + (−0.553 − 1.92i)4-s + (−0.834 − 0.481i)5-s + (−1.20 + 2.35i)7-s + (2.64 + 1.00i)8-s + (1.25 − 0.533i)10-s + (−4.74 + 2.74i)11-s + 3.75i·13-s + (−1.64 − 3.36i)14-s + (−3.38 + 2.12i)16-s + (0.594 − 0.343i)17-s + (−2.44 + 4.22i)19-s + (−0.464 + 1.87i)20-s + (0.941 − 7.69i)22-s + (1.07 + 0.620i)23-s + ⋯
L(s)  = 1  + (−0.601 + 0.798i)2-s + (−0.276 − 0.960i)4-s + (−0.373 − 0.215i)5-s + (−0.453 + 0.891i)7-s + (0.934 + 0.356i)8-s + (0.396 − 0.168i)10-s + (−1.43 + 0.826i)11-s + 1.04i·13-s + (−0.438 − 0.898i)14-s + (−0.846 + 0.531i)16-s + (0.144 − 0.0832i)17-s + (−0.560 + 0.969i)19-s + (−0.103 + 0.418i)20-s + (0.200 − 1.64i)22-s + (0.224 + 0.129i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.949 - 0.312i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0756592 + 0.472334i\)
\(L(\frac12)\) \(\approx\) \(0.0756592 + 0.472334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.850 - 1.12i)T \)
3 \( 1 \)
7 \( 1 + (1.20 - 2.35i)T \)
good5 \( 1 + (0.834 + 0.481i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.74 - 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.75iT - 13T^{2} \)
17 \( 1 + (-0.594 + 0.343i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.44 - 4.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.07 - 0.620i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 + (2.41 + 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.36 + 2.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.42iT - 41T^{2} \)
43 \( 1 - 5.97iT - 43T^{2} \)
47 \( 1 + (-1.80 + 3.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.04 + 3.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.34 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.01 - 5.20i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.17 + 4.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (5.76 - 3.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.543T + 83T^{2} \)
89 \( 1 + (0.480 + 0.277i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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

−12.51688607136125578516163691478, −11.46895960119696876472103076023, −10.21063145235761633013724605183, −9.536403664427981871567030621067, −8.449178605043851311272046310085, −7.70902473321198377719282215712, −6.54369935126180931561006730065, −5.52047122951286356214987331856, −4.39753247320784581755008745118, −2.23483402693866479893864198210, 0.44221263707881791532500924964, 2.77013762647804974879285367635, 3.71752356856237051760454098476, 5.23820421538997933461192126831, 6.98144419777817661452381272609, 7.84491894816587182215003795970, 8.690901874187948469099395223836, 10.01887494401690366855659916549, 10.68490873003058842450557938750, 11.23568432919293610300892338383

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