↑

Properties

Label 2-252-252.191-c1-0-17
Degree $2$
Conductor $252$
Sign $-0.669 - 0.742i$
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  + (0.571 + 1.29i)2-s + (1.27 + 1.17i)3-s + (−1.34 + 1.47i)4-s + 1.08i·5-s + (−0.786 + 2.31i)6-s + (−2.19 − 1.48i)7-s + (−2.68 − 0.894i)8-s + (0.252 + 2.98i)9-s + (−1.40 + 0.621i)10-s + 1.50·11-s + (−3.45 + 0.309i)12-s + (1.59 + 2.75i)13-s + (0.667 − 3.68i)14-s + (−1.27 + 1.38i)15-s + (−0.377 − 3.98i)16-s + (5.40 − 3.11i)17-s + ⋯
L(s)  = 1  + (0.404 + 0.914i)2-s + (0.736 + 0.676i)3-s + (−0.672 + 0.739i)4-s + 0.486i·5-s + (−0.321 + 0.947i)6-s + (−0.827 − 0.561i)7-s + (−0.948 − 0.316i)8-s + (0.0842 + 0.996i)9-s + (−0.444 + 0.196i)10-s + 0.453·11-s + (−0.996 + 0.0893i)12-s + (0.441 + 0.764i)13-s + (0.178 − 0.983i)14-s + (−0.328 + 0.357i)15-s + (−0.0943 − 0.995i)16-s + (1.31 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.669 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.675405 + 1.51853i\)
\(L(\frac12)\) \(\approx\) \(0.675405 + 1.51853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.571 - 1.29i)T \)
3 \( 1 + (-1.27 - 1.17i)T \)
7 \( 1 + (2.19 + 1.48i)T \)
good5 \( 1 - 1.08iT - 5T^{2} \)
11 \( 1 - 1.50T + 11T^{2} \)
13 \( 1 + (-1.59 - 2.75i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.40 + 3.11i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.29 + 1.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.25T + 23T^{2} \)
29 \( 1 + (1.78 + 1.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.708 + 0.409i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.35 - 7.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.57 + 3.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.95 + 2.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.81 + 11.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.686 - 0.396i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.19 + 2.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.36 + 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.96 - 4.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + (4.02 + 6.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.4 - 7.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.40 + 4.16i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.90 + 3.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.69 + 11.5i)T + (-48.5 - 84.0i)T^{2} \)
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

−12.74646546465579090163253298336, −11.44945402256247811354365995584, −10.18237213472424105236972169035, −9.380630032600264272372406521820, −8.508712098943514930407861862899, −7.27570547580559076383729998395, −6.58744631943248756110439969979, −5.12413236891335684295279911171, −3.89160955098092799459634858207, −3.08803877566653420347467000331, 1.27879748205287147393555037157, 2.88566643625006009175877147473, 3.79520626237876676825696032394, 5.51208408799567361911723694328, 6.45288300603771760262746156831, 8.065358784230053252372045418114, 8.956933122458956371850740159811, 9.675597738258333137022267584659, 10.81004628423131206576386807636, 12.10558174957650902965606248748

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