Properties

Label 2-252-63.20-c3-0-11
Degree $2$
Conductor $252$
Sign $0.155 + 0.987i$
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  + (−4.07 + 3.21i)3-s + (0.330 + 0.571i)5-s + (−15.6 + 9.91i)7-s + (6.26 − 26.2i)9-s + (21.4 + 12.3i)11-s + (−43.5 + 25.1i)13-s + (−3.18 − 1.26i)15-s − 67.5·17-s − 62.9i·19-s + (31.8 − 90.7i)21-s + (135. − 78.4i)23-s + (62.2 − 107. i)25-s + (58.9 + 127. i)27-s + (−129. − 74.9i)29-s + (139. − 80.5i)31-s + ⋯
L(s)  = 1  + (−0.784 + 0.619i)3-s + (0.0295 + 0.0511i)5-s + (−0.844 + 0.535i)7-s + (0.232 − 0.972i)9-s + (0.586 + 0.338i)11-s + (−0.928 + 0.536i)13-s + (−0.0548 − 0.0218i)15-s − 0.963·17-s − 0.760i·19-s + (0.331 − 0.943i)21-s + (1.23 − 0.710i)23-s + (0.498 − 0.863i)25-s + (0.420 + 0.907i)27-s + (−0.831 − 0.480i)29-s + (0.808 − 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.155 + 0.987i$
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: \((2,\ 252,\ (\ :3/2),\ 0.155 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5351783491\)
\(L(\frac12)\) \(\approx\) \(0.5351783491\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.07 - 3.21i)T \)
7 \( 1 + (15.6 - 9.91i)T \)
good5 \( 1 + (-0.330 - 0.571i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-21.4 - 12.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (43.5 - 25.1i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 67.5T + 4.91e3T^{2} \)
19 \( 1 + 62.9iT - 6.85e3T^{2} \)
23 \( 1 + (-135. + 78.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (129. + 74.9i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-139. + 80.5i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 16.2T + 5.06e4T^{2} \)
41 \( 1 + (134. + 233. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-188. + 325. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (31.3 - 54.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 136. iT - 1.48e5T^{2} \)
59 \( 1 + (358. + 621. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-23.9 - 13.8i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (163. + 283. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 246. iT - 3.57e5T^{2} \)
73 \( 1 + 261. iT - 3.89e5T^{2} \)
79 \( 1 + (391. - 678. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (599. - 1.03e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 968.T + 7.04e5T^{2} \)
97 \( 1 + (-1.10e3 - 639. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.41291906943811153811328611421, −10.43173815157155151623513191489, −9.439361467181955871859357617431, −8.909215729971475781274213521259, −6.97305491309792781892145973941, −6.42016731426787973437565568052, −5.09283265271576639107167509518, −4.13855306231793643215086422129, −2.55449782708942853835554599734, −0.25549331700906885804119166899, 1.24287234271101923221212472028, 3.09178850888428330414768438590, 4.66958280250006484430199742211, 5.85272171435387414199254607629, 6.84342488420454024116087935857, 7.54811628997177338268448050269, 8.982396293126953165104562943148, 10.04685904293755299251799528034, 10.96025734332655462986984235370, 11.81179123060469507912979987251

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