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Properties

Label 2-252-21.5-c3-0-2
Degree $2$
Conductor $252$
Sign $-0.262 - 0.964i$
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  + (4.27 − 7.39i)5-s + (−12.5 + 13.5i)7-s + (−27.4 + 15.8i)11-s + 9.92i·13-s + (63.7 + 110. i)17-s + (−100. − 58.2i)19-s + (55.8 + 32.2i)23-s + (26.0 + 45.0i)25-s + 113. i·29-s + (6.33 − 3.65i)31-s + (46.8 + 151. i)35-s + (−184. + 319. i)37-s + 211.·41-s − 432.·43-s + (−200. + 346. i)47-s + ⋯
L(s)  = 1  + (0.382 − 0.661i)5-s + (−0.679 + 0.733i)7-s + (−0.752 + 0.434i)11-s + 0.211i·13-s + (0.909 + 1.57i)17-s + (−1.21 − 0.703i)19-s + (0.506 + 0.292i)23-s + (0.208 + 0.360i)25-s + 0.723i·29-s + (0.0367 − 0.0211i)31-s + (0.226 + 0.729i)35-s + (−0.820 + 1.42i)37-s + 0.807·41-s − 1.53·43-s + (−0.620 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.262 - 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.067775835\)
\(L(\frac12)\) \(\approx\) \(1.067775835\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (12.5 - 13.5i)T \)
good5 \( 1 + (-4.27 + 7.39i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (27.4 - 15.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 9.92iT - 2.19e3T^{2} \)
17 \( 1 + (-63.7 - 110. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (100. + 58.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-55.8 - 32.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 113. iT - 2.43e4T^{2} \)
31 \( 1 + (-6.33 + 3.65i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (184. - 319. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 211.T + 6.89e4T^{2} \)
43 \( 1 + 432.T + 7.95e4T^{2} \)
47 \( 1 + (200. - 346. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (121. - 70.3i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-259. - 449. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-23.5 - 13.6i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-68.3 - 118. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 604. iT - 3.57e5T^{2} \)
73 \( 1 + (41.9 - 24.2i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-415. + 719. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 37.2T + 5.71e5T^{2} \)
89 \( 1 + (-235. + 407. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 522. iT - 9.12e5T^{2} \)
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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

−12.14379120291910975483947070063, −10.79654902037965191403603726497, −9.921188356817906746017482167270, −8.960431173516292571388970600373, −8.190084845544257197401624381724, −6.75494054244872751900421991658, −5.73609891378732783554807411375, −4.74693388810860215024188727861, −3.15210208683180229764810981857, −1.67796320238435111741742480310, 0.40910187322158419476926970646, 2.53148572080518187227480101440, 3.62704698615461002377012744603, 5.18732309613327039859258254311, 6.36719389643953439193479054052, 7.23251222295321138046926149175, 8.303365826876139263829684429585, 9.683824996862500372731079615088, 10.32542922760100278210786356444, 11.10313058354181207121292293352

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