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Properties

Label 2-252-63.25-c3-0-19
Degree $2$
Conductor $252$
Sign $-0.574 + 0.818i$
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  + (1.24 + 5.04i)3-s + (1.32 − 2.29i)5-s + (−14.0 + 12.0i)7-s + (−23.8 + 12.5i)9-s + (−25.1 − 43.5i)11-s + (−27.5 − 47.6i)13-s + (13.2 + 3.82i)15-s + (−11.3 + 19.7i)17-s + (9.43 + 16.3i)19-s + (−78.4 − 55.6i)21-s + (88.6 − 153. i)23-s + (58.9 + 102. i)25-s + (−93.2 − 104. i)27-s + (−122. + 211. i)29-s − 270.·31-s + ⋯
L(s)  = 1  + (0.240 + 0.970i)3-s + (0.118 − 0.205i)5-s + (−0.757 + 0.652i)7-s + (−0.884 + 0.466i)9-s + (−0.688 − 1.19i)11-s + (−0.587 − 1.01i)13-s + (0.228 + 0.0658i)15-s + (−0.162 + 0.281i)17-s + (0.113 + 0.197i)19-s + (−0.815 − 0.578i)21-s + (0.803 − 1.39i)23-s + (0.471 + 0.817i)25-s + (−0.664 − 0.746i)27-s + (−0.782 + 1.35i)29-s − 1.56·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.574 + 0.818i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1815356946\)
\(L(\frac12)\) \(\approx\) \(0.1815356946\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.24 - 5.04i)T \)
7 \( 1 + (14.0 - 12.0i)T \)
good5 \( 1 + (-1.32 + 2.29i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (25.1 + 43.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (27.5 + 47.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (11.3 - 19.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.43 - 16.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-88.6 + 153. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (122. - 211. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 270.T + 2.97e4T^{2} \)
37 \( 1 + (90.3 + 156. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (146. + 253. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-229. + 397. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 329.T + 1.03e5T^{2} \)
53 \( 1 + (295. - 512. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 23.5T + 2.05e5T^{2} \)
61 \( 1 + 861.T + 2.26e5T^{2} \)
67 \( 1 + 702.T + 3.00e5T^{2} \)
71 \( 1 - 99.3T + 3.57e5T^{2} \)
73 \( 1 + (439. - 761. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 113.T + 4.93e5T^{2} \)
83 \( 1 + (449. - 779. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-211. - 366. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-290. + 502. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.74822233007115215922133299611, −10.60030046582581730194165479656, −9.049113591857682304975310482949, −8.833029448549786238567034179258, −7.40977082515024067145809068085, −5.74061882237853424411678847891, −5.24425600048867495739766383144, −3.55239496631020472870476750050, −2.68717853516089064123428550357, −0.06449305936919593688807658871, 1.80609437961140497768460372859, 3.06129603933591273216115020940, 4.63162858951039453137167466316, 6.14797768068938187987159590594, 7.20472097957324176221973876369, 7.55711497677001070726627065063, 9.190836677642621794031548170149, 9.824521029754149513863172649091, 11.11066833637844164117621749794, 12.07270985611523782858508512012

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