↑

Properties

Label 2-252-252.95-c1-0-32
Degree $2$
Conductor $252$
Sign $0.999 - 0.0358i$
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  + (1.34 + 0.429i)2-s + (1.30 − 1.13i)3-s + (1.63 + 1.15i)4-s + 0.501i·5-s + (2.24 − 0.971i)6-s + (−2.46 − 0.949i)7-s + (1.70 + 2.25i)8-s + (0.414 − 2.97i)9-s + (−0.215 + 0.675i)10-s + 1.69·11-s + (3.44 − 0.342i)12-s + (−1.89 + 3.28i)13-s + (−2.92 − 2.33i)14-s + (0.570 + 0.655i)15-s + (1.32 + 3.77i)16-s + (−4.12 − 2.38i)17-s + ⋯
L(s)  = 1  + (0.952 + 0.303i)2-s + (0.754 − 0.656i)3-s + (0.815 + 0.578i)4-s + 0.224i·5-s + (0.918 − 0.396i)6-s + (−0.933 − 0.358i)7-s + (0.601 + 0.798i)8-s + (0.138 − 0.990i)9-s + (−0.0680 + 0.213i)10-s + 0.511·11-s + (0.995 − 0.0989i)12-s + (−0.526 + 0.911i)13-s + (−0.780 − 0.625i)14-s + (0.147 + 0.169i)15-s + (0.330 + 0.943i)16-s + (−1.00 − 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.999 - 0.0358i$
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: \((2,\ 252,\ (\ :1/2),\ 0.999 - 0.0358i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44127 + 0.0437995i\)
\(L(\frac12)\) \(\approx\) \(2.44127 + 0.0437995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.429i)T \)
3 \( 1 + (-1.30 + 1.13i)T \)
7 \( 1 + (2.46 + 0.949i)T \)
good5 \( 1 - 0.501iT - 5T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
13 \( 1 + (1.89 - 3.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.12 + 2.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0919 - 0.0530i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.55T + 23T^{2} \)
29 \( 1 + (-6.33 + 3.65i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.51 - 2.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.83 - 6.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.60 + 1.50i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.98 - 2.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.03 + 3.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.79 + 5.08i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.67 - 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.919 + 1.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.38 + 1.95i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.65T + 71T^{2} \)
73 \( 1 + (-5.55 + 9.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.07 - 2.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.75 + 11.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.73 + 2.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.45 - 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.24670221393791802648523499860, −11.56210254705758148376860920337, −10.09225126755024150679841114332, −9.019923036612239902102308992369, −7.85047901696406435173460459699, −6.64567612103904080351259865721, −6.52604010503642798533859685769, −4.53721214078509464840557852315, −3.43178662807264187611893571875, −2.23582036855190262695378708269, 2.34515787803616669354444359933, 3.46571027501604942227378841252, 4.49789218050686137471053597671, 5.70690502326655676492126017381, 6.83985782865327136842331367442, 8.233175748341802937182289019565, 9.380036841618006705157400003784, 10.17346931373836913532705371778, 11.03569763484893525089547060073, 12.40970585369555411519531538375

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