↑

Properties

Label 2-252-63.38-c1-0-6
Degree $2$
Conductor $252$
Sign $0.110 + 0.993i$
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  + (0.106 − 1.72i)3-s + (1.43 − 2.48i)5-s + (2.56 + 0.632i)7-s + (−2.97 − 0.369i)9-s + (−2.34 + 1.35i)11-s + (−3.18 + 1.84i)13-s + (−4.14 − 2.74i)15-s + (3.22 − 5.58i)17-s + (2.73 − 1.58i)19-s + (1.36 − 4.37i)21-s + (2.59 + 1.49i)23-s + (−1.61 − 2.79i)25-s + (−0.956 + 5.10i)27-s + (−2.48 − 1.43i)29-s + 9.54i·31-s + ⋯
L(s)  = 1  + (0.0616 − 0.998i)3-s + (0.641 − 1.11i)5-s + (0.970 + 0.239i)7-s + (−0.992 − 0.123i)9-s + (−0.708 + 0.408i)11-s + (−0.884 + 0.510i)13-s + (−1.06 − 0.708i)15-s + (0.781 − 1.35i)17-s + (0.628 − 0.362i)19-s + (0.298 − 0.954i)21-s + (0.540 + 0.311i)23-s + (−0.322 − 0.558i)25-s + (−0.184 + 0.982i)27-s + (−0.461 − 0.266i)29-s + 1.71i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.110 + 0.993i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.110 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05079 - 0.940213i\)
\(L(\frac12)\) \(\approx\) \(1.05079 - 0.940213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.106 + 1.72i)T \)
7 \( 1 + (-2.56 - 0.632i)T \)
good5 \( 1 + (-1.43 + 2.48i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.34 - 1.35i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.18 - 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.22 + 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.73 + 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.48 + 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.54iT - 31T^{2} \)
37 \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.794 + 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + (-2.16 - 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.67T + 59T^{2} \)
61 \( 1 - 0.654iT - 61T^{2} \)
67 \( 1 - 7.72T + 67T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.19T + 79T^{2} \)
83 \( 1 + (7.92 - 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.14 - 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.2 + 7.62i)T + (48.5 + 84.0i)T^{2} \)
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

−12.13197678831594721028840611730, −11.13017455222331665781798627628, −9.623445417062445072827687085240, −8.901512172756164405737981355399, −7.80444495579725646697908232875, −7.06651723984991895181516203607, −5.32462897431655010293301045442, −5.04496041091278908293324703768, −2.58497559883463327342411115046, −1.29737058603080776079498180222, 2.43966743541553558925141968895, 3.66784011514598248901920485500, 5.14520462228100217908997001648, 5.92495931897167343205706267538, 7.48741782365060959627605578071, 8.342458141405843796092846321975, 9.720789198475316511470143105883, 10.47199794780840372236132401331, 10.88803340522592664781094309784, 12.03732317788832636799886968905

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