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Properties

Label 2-252-63.20-c3-0-20
Degree $2$
Conductor $252$
Sign $-0.990 + 0.137i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.494 − 5.17i)3-s + (−2.99 − 5.19i)5-s + (16.2 − 8.93i)7-s + (−26.5 − 5.11i)9-s + (−39.3 − 22.7i)11-s + (22.7 − 13.1i)13-s + (−28.3 + 12.9i)15-s + 19.7·17-s + 27.9i·19-s + (−38.1 − 88.3i)21-s + (−60.3 + 34.8i)23-s + (44.5 − 77.0i)25-s + (−39.5 + 134. i)27-s + (−119. − 68.7i)29-s + (−138. + 79.7i)31-s + ⋯
L(s)  = 1  + (0.0951 − 0.995i)3-s + (−0.268 − 0.464i)5-s + (0.875 − 0.482i)7-s + (−0.981 − 0.189i)9-s + (−1.07 − 0.623i)11-s + (0.485 − 0.280i)13-s + (−0.488 + 0.222i)15-s + 0.281·17-s + 0.337i·19-s + (−0.396 − 0.917i)21-s + (−0.547 + 0.315i)23-s + (0.356 − 0.616i)25-s + (−0.282 + 0.959i)27-s + (−0.762 − 0.440i)29-s + (−0.800 + 0.462i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.990 + 0.137i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.990 + 0.137i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.197524102\)
\(L(\frac12)\) \(\approx\) \(1.197524102\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.494 + 5.17i)T \)
7 \( 1 + (-16.2 + 8.93i)T \)
good5 \( 1 + (2.99 + 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (39.3 + 22.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22.7 + 13.1i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 19.7T + 4.91e3T^{2} \)
19 \( 1 - 27.9iT - 6.85e3T^{2} \)
23 \( 1 + (60.3 - 34.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (119. + 68.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (138. - 79.7i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 287.T + 5.06e4T^{2} \)
41 \( 1 + (-20.4 - 35.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-55.4 + 95.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (109. - 189. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 209. iT - 1.48e5T^{2} \)
59 \( 1 + (413. + 716. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-594. - 343. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (171. + 296. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 387. iT - 3.57e5T^{2} \)
73 \( 1 + 220. iT - 3.89e5T^{2} \)
79 \( 1 + (-242. + 419. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-354. + 613. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 140.T + 7.04e5T^{2} \)
97 \( 1 + (-1.30e3 - 753. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.22919376624789719292710333451, −10.44018242634903984134655522917, −8.820735411562986768072324956115, −8.045263958764651512873410033152, −7.46691394411705138035921572617, −6.02053777962715363441751345172, −5.04021990271286805303704154345, −3.44500977658345448057666922162, −1.81945897003559035053896827584, −0.45443083067448152624076093370, 2.22273875773079305301532669526, 3.58250240723554691140663795735, 4.83591404831642016654266997061, 5.65112181436574829033791470822, 7.27209207782578104499565411668, 8.274666846097586731655883756491, 9.174356039863596693933518108730, 10.30398911624541739758090252918, 10.99846198116251400272639926630, 11.75262915487627859423507241187

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