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Properties

Label 2-252-63.59-c3-0-7
Degree $2$
Conductor $252$
Sign $-0.589 - 0.808i$
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  + (−1.46 + 4.98i)3-s − 2.27·5-s + (17.2 − 6.82i)7-s + (−22.6 − 14.6i)9-s + 35.8i·11-s + (70.2 + 40.5i)13-s + (3.33 − 11.3i)15-s + (−25.8 + 44.7i)17-s + (−9.61 + 5.55i)19-s + (8.76 + 95.8i)21-s − 30.5i·23-s − 119.·25-s + (106. − 91.7i)27-s + (−167. + 96.9i)29-s + (−162. + 94.0i)31-s + ⋯
L(s)  = 1  + (−0.282 + 0.959i)3-s − 0.203·5-s + (0.929 − 0.368i)7-s + (−0.840 − 0.541i)9-s + 0.983i·11-s + (1.49 + 0.865i)13-s + (0.0574 − 0.195i)15-s + (−0.368 + 0.638i)17-s + (−0.116 + 0.0670i)19-s + (0.0910 + 0.995i)21-s − 0.277i·23-s − 0.958·25-s + (0.756 − 0.653i)27-s + (−1.07 + 0.620i)29-s + (−0.943 + 0.544i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 - 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.589 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.589 - 0.808i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.589 - 0.808i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.386397722\)
\(L(\frac12)\) \(\approx\) \(1.386397722\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.46 - 4.98i)T \)
7 \( 1 + (-17.2 + 6.82i)T \)
good5 \( 1 + 2.27T + 125T^{2} \)
11 \( 1 - 35.8iT - 1.33e3T^{2} \)
13 \( 1 + (-70.2 - 40.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (25.8 - 44.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (9.61 - 5.55i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 30.5iT - 1.21e4T^{2} \)
29 \( 1 + (167. - 96.9i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (162. - 94.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-25.3 - 43.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (151. - 261. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-28.1 - 48.7i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (54.1 - 93.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-548. - 316. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (343. + 594. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-340. - 196. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (9.43 + 16.3i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.11e3iT - 3.57e5T^{2} \)
73 \( 1 + (-458. - 264. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (231. - 401. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (212. + 368. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (74.5 + 129. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.05e3 + 611. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.52513025402110957198592527546, −11.08788109051081633932939617788, −10.14640229512341208818679098135, −9.058634337199155350349666341399, −8.228771611169356051788024578120, −6.90145437361751249072031514882, −5.67187300047442103592500995014, −4.45435087363362334231171751079, −3.78171686195041090771803737123, −1.67577304936730224515415758207, 0.58524398064970976614323268016, 2.02402114328786750821643860418, 3.62479817737085245992689034751, 5.41736314709090233326270982169, 6.02524381728988966756199814800, 7.44289175896801832574603937929, 8.220187841532305622731579538292, 8.979977271228122604033562926925, 10.78690257242833132392809941546, 11.30892689264250643137270567827

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