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Properties

Label 2-252-63.59-c3-0-2
Degree $2$
Conductor $252$
Sign $0.522 - 0.852i$
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  + (−4.84 + 1.86i)3-s − 21.0·5-s + (9.48 − 15.9i)7-s + (20.0 − 18.0i)9-s − 33.5i·11-s + (−38.7 − 22.3i)13-s + (102. − 39.2i)15-s + (−60.7 + 105. i)17-s + (11.4 − 6.59i)19-s + (−16.3 + 94.8i)21-s + 183. i·23-s + 317.·25-s + (−63.4 + 125. i)27-s + (17.1 − 9.91i)29-s + (6.16 − 3.56i)31-s + ⋯
L(s)  = 1  + (−0.933 + 0.359i)3-s − 1.88·5-s + (0.512 − 0.858i)7-s + (0.742 − 0.670i)9-s − 0.920i·11-s + (−0.825 − 0.476i)13-s + (1.75 − 0.675i)15-s + (−0.867 + 1.50i)17-s + (0.138 − 0.0796i)19-s + (−0.169 + 0.985i)21-s + 1.66i·23-s + 2.54·25-s + (−0.451 + 0.892i)27-s + (0.109 − 0.0634i)29-s + (0.0357 − 0.0206i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.522 - 0.852i$
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: \((2,\ 252,\ (\ :3/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5500995671\)
\(L(\frac12)\) \(\approx\) \(0.5500995671\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.84 - 1.86i)T \)
7 \( 1 + (-9.48 + 15.9i)T \)
good5 \( 1 + 21.0T + 125T^{2} \)
11 \( 1 + 33.5iT - 1.33e3T^{2} \)
13 \( 1 + (38.7 + 22.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (60.7 - 105. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-11.4 + 6.59i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 183. iT - 1.21e4T^{2} \)
29 \( 1 + (-17.1 + 9.91i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-6.16 + 3.56i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-82.0 - 142. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-181. + 315. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (86.9 + 150. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-78.1 + 135. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-133. - 76.9i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-389. - 674. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-141. - 81.4i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-58.7 - 101. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 163. iT - 3.57e5T^{2} \)
73 \( 1 + (-711. - 411. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (275. - 477. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (188. + 326. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-35.7 - 61.8i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.17e3 + 677. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.53452163553362770243160453058, −11.02798269225137826881269366746, −10.24797535954590929396615430623, −8.649819621256721525449379533267, −7.71055968046151979023644954411, −6.94218752104565128989168188284, −5.45198689435079410729671090456, −4.25119775046454634807017123702, −3.64056015343215757312095799950, −0.806500091546734311402195657789, 0.37998174910996851740598929911, 2.45469966262598626254283264635, 4.51283651753182109221430796862, 4.84796745104104657684752072286, 6.66198891739313378370957019754, 7.39271423280088925340476446939, 8.261934591830980418882474819781, 9.513746826540918202356700370655, 10.98614771606632390425368665304, 11.55301274407132167398442473084

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