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Properties

Label 2-252-9.4-c3-0-10
Degree $2$
Conductor $252$
Sign $0.953 - 0.300i$
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  + (5.09 + 1.01i)3-s + (0.184 − 0.319i)5-s + (−3.5 − 6.06i)7-s + (24.9 + 10.3i)9-s + (16.1 + 27.9i)11-s + (4.62 − 8.01i)13-s + (1.26 − 1.44i)15-s + 87.0·17-s + 66.5·19-s + (−11.6 − 34.4i)21-s + (21.4 − 37.1i)23-s + (62.4 + 108. i)25-s + (116. + 77.9i)27-s + (−48.3 − 83.7i)29-s + (−11.7 + 20.3i)31-s + ⋯
L(s)  = 1  + (0.980 + 0.195i)3-s + (0.0164 − 0.0285i)5-s + (−0.188 − 0.327i)7-s + (0.923 + 0.382i)9-s + (0.441 + 0.764i)11-s + (0.0987 − 0.170i)13-s + (0.0217 − 0.0248i)15-s + 1.24·17-s + 0.803·19-s + (−0.121 − 0.357i)21-s + (0.194 − 0.336i)23-s + (0.499 + 0.865i)25-s + (0.831 + 0.555i)27-s + (−0.309 − 0.536i)29-s + (−0.0679 + 0.117i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.953 - 0.300i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.953 - 0.300i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.704755489\)
\(L(\frac12)\) \(\approx\) \(2.704755489\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.09 - 1.01i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good5 \( 1 + (-0.184 + 0.319i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-16.1 - 27.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-4.62 + 8.01i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 87.0T + 4.91e3T^{2} \)
19 \( 1 - 66.5T + 6.85e3T^{2} \)
23 \( 1 + (-21.4 + 37.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (48.3 + 83.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (11.7 - 20.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 147.T + 5.06e4T^{2} \)
41 \( 1 + (12.7 - 22.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-11.0 - 19.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (54.6 + 94.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 + (-194. + 336. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-183. - 318. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (239. - 415. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 186.T + 3.57e5T^{2} \)
73 \( 1 + 722.T + 3.89e5T^{2} \)
79 \( 1 + (87.2 + 151. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (455. + 788. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 592.T + 7.04e5T^{2} \)
97 \( 1 + (604. + 1.04e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.73486165975256259500584730199, −10.37222284160337428297657354406, −9.724219714012865889386654971761, −8.819749063164801243827866978922, −7.68271998376362758905955565584, −6.96791915474587221574877631486, −5.33960879003953665309710413261, −4.04631205670476615201490698968, −3.00589099038655999974194110082, −1.40321733824298747056156922464, 1.22665566674556144631949586868, 2.85680812032729565296049437432, 3.79925626134470170780124488939, 5.42484339436057826257718132268, 6.67861224094820428066806517734, 7.73239793756570823958560762868, 8.677715875328820958173768454277, 9.454207819497239123242064285607, 10.42679161044357671034256330026, 11.73929645352143908552906280405

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