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Properties

Label 2-252-63.47-c3-0-1
Degree $2$
Conductor $252$
Sign $0.305 - 0.952i$
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  + (−3.93 − 3.39i)3-s − 12.5·5-s + (−6.65 − 17.2i)7-s + (3.94 + 26.7i)9-s − 68.7i·11-s + (−34.1 + 19.7i)13-s + (49.4 + 42.7i)15-s + (38.3 + 66.4i)17-s + (−69.3 − 40.0i)19-s + (−32.5 + 90.5i)21-s + 155. i·23-s + 33.3·25-s + (75.1 − 118. i)27-s + (197. + 114. i)29-s + (126. + 72.7i)31-s + ⋯
L(s)  = 1  + (−0.756 − 0.653i)3-s − 1.12·5-s + (−0.359 − 0.933i)7-s + (0.145 + 0.989i)9-s − 1.88i·11-s + (−0.728 + 0.420i)13-s + (0.851 + 0.735i)15-s + (0.547 + 0.948i)17-s + (−0.836 − 0.483i)19-s + (−0.337 + 0.941i)21-s + 1.40i·23-s + 0.266·25-s + (0.535 − 0.844i)27-s + (1.26 + 0.731i)29-s + (0.730 + 0.421i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.305 - 0.952i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2872224299\)
\(L(\frac12)\) \(\approx\) \(0.2872224299\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.93 + 3.39i)T \)
7 \( 1 + (6.65 + 17.2i)T \)
good5 \( 1 + 12.5T + 125T^{2} \)
11 \( 1 + 68.7iT - 1.33e3T^{2} \)
13 \( 1 + (34.1 - 19.7i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-38.3 - 66.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.3 + 40.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 155. iT - 1.21e4T^{2} \)
29 \( 1 + (-197. - 114. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-126. - 72.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-74.1 + 128. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (27.9 + 48.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (109. - 188. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-54.9 - 95.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-257. + 148. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (89.5 - 155. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (275. - 159. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-326. + 566. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 157. iT - 3.57e5T^{2} \)
73 \( 1 + (1.05e3 - 609. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (409. + 709. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (404. - 700. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (20.9 - 36.3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-161. - 93.1i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.68182320732128665551357059918, −11.05664349533684135557469338086, −10.22988098426447025746889457240, −8.561176639505990247124268476567, −7.75502642064414140934643487013, −6.84751639817047930131414531158, −5.85435279911361033358230171936, −4.42889964493566611029663846104, −3.26240932453150730885473892302, −1.01470562117128761973087952192, 0.15769136195233376925697628070, 2.66124524497023629195977103542, 4.27803132020305190494674152421, 4.93092728156882401602706575485, 6.33125121760810890850681665659, 7.36326761202280332190516519838, 8.522934950404503796719362314602, 9.809554740335655284109365393838, 10.25280631532053658424923505923, 11.77490458628374832296288600002

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