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Properties

Label 2-252-63.5-c3-0-12
Degree $2$
Conductor $252$
Sign $0.995 - 0.0925i$
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  + (5.18 − 0.383i)3-s + (−3.01 − 5.22i)5-s + (7.56 + 16.9i)7-s + (26.7 − 3.97i)9-s + (−15.9 − 9.20i)11-s + (55.8 + 32.2i)13-s + (−17.6 − 25.9i)15-s + (8.85 + 15.3i)17-s + (48.2 + 27.8i)19-s + (45.6 + 84.6i)21-s + (132. − 76.4i)23-s + (44.2 − 76.6i)25-s + (136. − 30.8i)27-s + (−111. + 64.4i)29-s − 71.5i·31-s + ⋯
L(s)  = 1  + (0.997 − 0.0737i)3-s + (−0.270 − 0.467i)5-s + (0.408 + 0.912i)7-s + (0.989 − 0.147i)9-s + (−0.437 − 0.252i)11-s + (1.19 + 0.687i)13-s + (−0.303 − 0.446i)15-s + (0.126 + 0.218i)17-s + (0.582 + 0.336i)19-s + (0.474 + 0.880i)21-s + (1.19 − 0.692i)23-s + (0.354 − 0.613i)25-s + (0.975 − 0.219i)27-s + (−0.714 + 0.412i)29-s − 0.414i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.995 - 0.0925i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.995 - 0.0925i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.705400845\)
\(L(\frac12)\) \(\approx\) \(2.705400845\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.18 + 0.383i)T \)
7 \( 1 + (-7.56 - 16.9i)T \)
good5 \( 1 + (3.01 + 5.22i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (15.9 + 9.20i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-55.8 - 32.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-8.85 - 15.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-48.2 - 27.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-132. + 76.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (111. - 64.4i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 71.5iT - 2.97e4T^{2} \)
37 \( 1 + (62.1 - 107. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-29.4 + 50.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-39.6 - 68.7i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 161.T + 1.03e5T^{2} \)
53 \( 1 + (-347. + 200. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 94.0T + 2.05e5T^{2} \)
61 \( 1 - 255. iT - 2.26e5T^{2} \)
67 \( 1 + 107.T + 3.00e5T^{2} \)
71 \( 1 - 234. iT - 3.57e5T^{2} \)
73 \( 1 + (-578. + 334. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 694.T + 4.93e5T^{2} \)
83 \( 1 + (542. + 938. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (813. - 1.40e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.57e3 - 907. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.70512013571002471083585453760, −10.66838292310723204000797250747, −9.341307394218131239714919237777, −8.636007518139951406193885936876, −8.044566448595072231543554632194, −6.71484041730657231879228043767, −5.34870810969769266355223612125, −4.09442470399140086508783086169, −2.80790912493589560468905650643, −1.38929497407832323813798771835, 1.24212802673988313491961767573, 3.00498003624137310074106925012, 3.88029248432989790518273862838, 5.25179723653655393258207122390, 7.01715311410774222181647805214, 7.59779739081451234195746085843, 8.578313028847731344088009921034, 9.654934203206798758644054560958, 10.67714096137873756948280209286, 11.28266649829704324845328902677

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