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Properties

Label 2-252-63.41-c3-0-12
Degree $2$
Conductor $252$
Sign $0.938 - 0.344i$
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  + (4.09 − 3.19i)3-s + (−3.53 + 6.12i)5-s + (7.59 + 16.8i)7-s + (6.56 − 26.1i)9-s + (7.40 − 4.27i)11-s + (45.3 + 26.1i)13-s + (5.08 + 36.3i)15-s − 38.9·17-s + 66.4i·19-s + (85.1 + 44.9i)21-s + (173. + 100. i)23-s + (37.5 + 64.9i)25-s + (−56.7 − 128. i)27-s + (52.9 − 30.5i)29-s + (−116. − 67.2i)31-s + ⋯
L(s)  = 1  + (0.788 − 0.615i)3-s + (−0.316 + 0.547i)5-s + (0.410 + 0.912i)7-s + (0.243 − 0.969i)9-s + (0.202 − 0.117i)11-s + (0.967 + 0.558i)13-s + (0.0875 + 0.626i)15-s − 0.555·17-s + 0.801i·19-s + (0.884 + 0.466i)21-s + (1.57 + 0.910i)23-s + (0.300 + 0.519i)25-s + (−0.404 − 0.914i)27-s + (0.339 − 0.195i)29-s + (−0.674 − 0.389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.422433971\)
\(L(\frac12)\) \(\approx\) \(2.422433971\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.09 + 3.19i)T \)
7 \( 1 + (-7.59 - 16.8i)T \)
good5 \( 1 + (3.53 - 6.12i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7.40 + 4.27i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-45.3 - 26.1i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 38.9T + 4.91e3T^{2} \)
19 \( 1 - 66.4iT - 6.85e3T^{2} \)
23 \( 1 + (-173. - 100. i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-52.9 + 30.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (116. + 67.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 298.T + 5.06e4T^{2} \)
41 \( 1 + (-221. + 383. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-26.1 - 45.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (137. + 238. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 136. iT - 1.48e5T^{2} \)
59 \( 1 + (191. - 331. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (261. - 151. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-318. + 552. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 228. iT - 3.57e5T^{2} \)
73 \( 1 - 1.24e3iT - 3.89e5T^{2} \)
79 \( 1 + (100. + 174. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (323. + 560. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 826.T + 7.04e5T^{2} \)
97 \( 1 + (-17.0 + 9.85i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.60698476351785811891450832056, −10.97702783596082877891483481795, −9.346554160517510571373506421010, −8.777112569312527517553706355202, −7.74509523854565740721961554665, −6.79432245477421857238836381428, −5.71774622600248876323614468274, −3.96930393527748122020887878805, −2.81732798725772340949102685523, −1.48812880456432851080236141030, 1.03152435422959753923672781415, 2.92603113223569878780778281765, 4.23154065540545566961505178989, 4.89877728578085851424934580268, 6.68743063219829567105602656972, 7.88184399419129696189564185713, 8.621982681214666187114380593818, 9.476588048627646355789500240760, 10.76597448331309193728238377956, 11.12596679956129071683060833776

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