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Properties

Label 2-252-63.47-c3-0-21
Degree $2$
Conductor $252$
Sign $0.0403 + 0.999i$
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  + (2.56 − 4.51i)3-s + 17.7·5-s + (−5.94 − 17.5i)7-s + (−13.8 − 23.1i)9-s + 13.5i·11-s + (45.1 − 26.0i)13-s + (45.6 − 80.4i)15-s + (−6.56 − 11.3i)17-s + (−53.6 − 30.9i)19-s + (−94.5 − 18.1i)21-s + 151. i·23-s + 191.·25-s + (−140. + 3.15i)27-s + (133. + 76.8i)29-s + (−83.7 − 48.3i)31-s + ⋯
L(s)  = 1  + (0.493 − 0.869i)3-s + 1.59·5-s + (−0.320 − 0.947i)7-s + (−0.512 − 0.858i)9-s + 0.371i·11-s + (0.962 − 0.555i)13-s + (0.785 − 1.38i)15-s + (−0.0937 − 0.162i)17-s + (−0.648 − 0.374i)19-s + (−0.982 − 0.188i)21-s + 1.37i·23-s + 1.53·25-s + (−0.999 + 0.0225i)27-s + (0.852 + 0.492i)29-s + (−0.485 − 0.280i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0403 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0403 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.0403 + 0.999i$
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: \((2,\ 252,\ (\ :3/2),\ 0.0403 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.635529250\)
\(L(\frac12)\) \(\approx\) \(2.635529250\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.56 + 4.51i)T \)
7 \( 1 + (5.94 + 17.5i)T \)
good5 \( 1 - 17.7T + 125T^{2} \)
11 \( 1 - 13.5iT - 1.33e3T^{2} \)
13 \( 1 + (-45.1 + 26.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (6.56 + 11.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (53.6 + 30.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 - 151. iT - 1.21e4T^{2} \)
29 \( 1 + (-133. - 76.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (83.7 + 48.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-135. + 235. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (171. + 296. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (127. - 221. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (15.5 + 26.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-324. + 187. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-331. + 574. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-530. + 306. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (298. - 517. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 265. iT - 3.57e5T^{2} \)
73 \( 1 + (886. - 511. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-582. - 1.00e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-177. + 306. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (677. - 1.17e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-326. - 188. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.30730811981386413187438810658, −10.24973368398169197614001749544, −9.462186359870149750272404293119, −8.486654531038650599123402864105, −7.22283736502433242099415628082, −6.43992514357613939350983048112, −5.46367989179193324596705261434, −3.61247001317158653948211307513, −2.21336511720983528485914691260, −1.04193029423573936625136582328, 1.94616476204419376816066849140, 3.00122792978232007940709565092, 4.57486852621038638240104945752, 5.81709312719555547280567600029, 6.39869586262473416276125394708, 8.554820697655310147743172211528, 8.850595484744122539343836081029, 9.957450510942633006851659810863, 10.50475189427562259592392771326, 11.73116914850312778509747834672

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