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Properties

Label 2-252-63.59-c3-0-16
Degree $2$
Conductor $252$
Sign $-0.832 + 0.553i$
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  + (−5.09 − 1.04i)3-s − 1.90·5-s + (3.04 + 18.2i)7-s + (24.8 + 10.6i)9-s − 38.8i·11-s + (8.40 + 4.85i)13-s + (9.67 + 1.97i)15-s + (−22.9 + 39.6i)17-s + (19.7 − 11.4i)19-s + (3.49 − 96.1i)21-s − 156. i·23-s − 121.·25-s + (−115. − 79.8i)27-s + (−187. + 108. i)29-s + (−174. + 100. i)31-s + ⋯
L(s)  = 1  + (−0.979 − 0.200i)3-s − 0.169·5-s + (0.164 + 0.986i)7-s + (0.919 + 0.392i)9-s − 1.06i·11-s + (0.179 + 0.103i)13-s + (0.166 + 0.0340i)15-s + (−0.326 + 0.565i)17-s + (0.239 − 0.138i)19-s + (0.0363 − 0.999i)21-s − 1.42i·23-s − 0.971·25-s + (−0.822 − 0.568i)27-s + (−1.19 + 0.692i)29-s + (−1.00 + 0.582i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.832 + 0.553i$
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: \((2,\ 252,\ (\ :3/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3259889306\)
\(L(\frac12)\) \(\approx\) \(0.3259889306\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.09 + 1.04i)T \)
7 \( 1 + (-3.04 - 18.2i)T \)
good5 \( 1 + 1.90T + 125T^{2} \)
11 \( 1 + 38.8iT - 1.33e3T^{2} \)
13 \( 1 + (-8.40 - 4.85i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (22.9 - 39.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-19.7 + 11.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 156. iT - 1.21e4T^{2} \)
29 \( 1 + (187. - 108. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (174. - 100. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (146. + 253. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-119. + 206. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (158. + 275. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-76.4 + 132. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-26.3 - 15.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (233. + 404. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-439. - 253. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (315. + 546. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 413. iT - 3.57e5T^{2} \)
73 \( 1 + (-276. - 159. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (502. - 870. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (415. + 719. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-572. - 990. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.55e3 - 899. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.17248314741275452477256520773, −10.64136318558419203599740998732, −9.190954163182104697337177168698, −8.329817741265384885396732653097, −7.04861775627951044555505174094, −5.94802715687878407922892867707, −5.28039759686765783629205320040, −3.78073257555201731648355776754, −1.96057731666962045645807993942, −0.14832500652789629879604350140, 1.51193794310523599476447807527, 3.77442631635276740063910734959, 4.70291587906393080517862291958, 5.84290181039976395790543007087, 7.12404952936082034352816736621, 7.68855422477658082892877983902, 9.524287851675984348590065330069, 10.04069586916696362932716325952, 11.25162644967513866219923272508, 11.66467683101514478652790060234

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