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Properties

Label 2-252-28.27-c3-0-34
Degree $2$
Conductor $252$
Sign $0.906 - 0.422i$
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  + (−0.414 + 2.79i)2-s + (−7.65 − 2.31i)4-s + 8.74i·5-s + (13.8 − 12.3i)7-s + (9.65 − 20.4i)8-s + (−24.4 − 3.62i)10-s + 0.397i·11-s − 75.7i·13-s + (28.8 + 43.7i)14-s + (53.2 + 35.4i)16-s − 84.4i·17-s − 20.0·19-s + (20.2 − 66.9i)20-s + (−1.11 − 0.164i)22-s + 122. i·23-s + ⋯
L(s)  = 1  + (−0.146 + 0.989i)2-s + (−0.957 − 0.289i)4-s + 0.782i·5-s + (0.745 − 0.666i)7-s + (0.426 − 0.904i)8-s + (−0.773 − 0.114i)10-s + 0.0109i·11-s − 1.61i·13-s + (0.550 + 0.834i)14-s + (0.832 + 0.554i)16-s − 1.20i·17-s − 0.242·19-s + (0.226 − 0.748i)20-s + (−0.0107 − 0.00159i)22-s + 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.906 - 0.422i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.542404238\)
\(L(\frac12)\) \(\approx\) \(1.542404238\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.414 - 2.79i)T \)
3 \( 1 \)
7 \( 1 + (-13.8 + 12.3i)T \)
good5 \( 1 - 8.74iT - 125T^{2} \)
11 \( 1 - 0.397iT - 1.33e3T^{2} \)
13 \( 1 + 75.7iT - 2.19e3T^{2} \)
17 \( 1 + 84.4iT - 4.91e3T^{2} \)
19 \( 1 + 20.0T + 6.85e3T^{2} \)
23 \( 1 - 122. iT - 1.21e4T^{2} \)
29 \( 1 - 175.T + 2.43e4T^{2} \)
31 \( 1 - 128.T + 2.97e4T^{2} \)
37 \( 1 - 253.T + 5.06e4T^{2} \)
41 \( 1 + 81.4iT - 6.89e4T^{2} \)
43 \( 1 + 69.1iT - 7.95e4T^{2} \)
47 \( 1 + 147.T + 1.03e5T^{2} \)
53 \( 1 + 283.T + 1.48e5T^{2} \)
59 \( 1 - 632.T + 2.05e5T^{2} \)
61 \( 1 + 3.25iT - 2.26e5T^{2} \)
67 \( 1 - 551. iT - 3.00e5T^{2} \)
71 \( 1 + 486. iT - 3.57e5T^{2} \)
73 \( 1 + 165. iT - 3.89e5T^{2} \)
79 \( 1 - 279. iT - 4.93e5T^{2} \)
83 \( 1 + 622.T + 5.71e5T^{2} \)
89 \( 1 + 696. iT - 7.04e5T^{2} \)
97 \( 1 + 1.62e3iT - 9.12e5T^{2} \)
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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

−11.49494741109002343831617955114, −10.49674907525708287622162174443, −9.807726389418973159692727947352, −8.410301093213514404222987985801, −7.62251679907672853264815410502, −6.86447584549021593466494632992, −5.61242327630946442558585801377, −4.60100312821744041496756926978, −3.10921794619692674823756521692, −0.78164187559715192905087243688, 1.25524335490926524797846955332, 2.39875736404682881069508212393, 4.22767280869453675026258740036, 4.87647907383505725749952030025, 6.36202793719095562185519665600, 8.205513874846921011549805680579, 8.655398077543267956909627954627, 9.582237407312466453503745689636, 10.72842671361538126483655600927, 11.62570474981312331200349682767

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