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Properties

Label 2-252-63.41-c3-0-13
Degree $2$
Conductor $252$
Sign $0.406 + 0.913i$
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  + (−2.29 + 4.65i)3-s + (−6.03 + 10.4i)5-s + (−16.9 − 7.37i)7-s + (−16.4 − 21.4i)9-s + (0.00221 − 0.00127i)11-s + (−6.06 − 3.50i)13-s + (−34.8 − 52.1i)15-s + 28.3·17-s − 49.1i·19-s + (73.4 − 62.1i)21-s + (44.4 + 25.6i)23-s + (−10.3 − 17.9i)25-s + (137. − 27.2i)27-s + (97.9 − 56.5i)29-s + (28.4 + 16.4i)31-s + ⋯
L(s)  = 1  + (−0.442 + 0.896i)3-s + (−0.539 + 0.935i)5-s + (−0.917 − 0.398i)7-s + (−0.608 − 0.793i)9-s + (6.06e−5 − 3.50e−5i)11-s + (−0.129 − 0.0747i)13-s + (−0.599 − 0.897i)15-s + 0.403·17-s − 0.594i·19-s + (0.763 − 0.646i)21-s + (0.403 + 0.232i)23-s + (−0.0828 − 0.143i)25-s + (0.980 − 0.194i)27-s + (0.627 − 0.362i)29-s + (0.164 + 0.0950i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.406 + 0.913i$
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: \((2,\ 252,\ (\ :3/2),\ 0.406 + 0.913i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4562171388\)
\(L(\frac12)\) \(\approx\) \(0.4562171388\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.29 - 4.65i)T \)
7 \( 1 + (16.9 + 7.37i)T \)
good5 \( 1 + (6.03 - 10.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-0.00221 + 0.00127i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (6.06 + 3.50i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 28.3T + 4.91e3T^{2} \)
19 \( 1 + 49.1iT - 6.85e3T^{2} \)
23 \( 1 + (-44.4 - 25.6i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-97.9 + 56.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-28.4 - 16.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 101.T + 5.06e4T^{2} \)
41 \( 1 + (-11.2 + 19.4i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (227. + 393. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (231. + 400. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 567. iT - 1.48e5T^{2} \)
59 \( 1 + (-145. + 252. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (592. - 341. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-269. + 467. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 307. iT - 3.57e5T^{2} \)
73 \( 1 - 495. iT - 3.89e5T^{2} \)
79 \( 1 + (324. + 562. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (565. + 979. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 130.T + 7.04e5T^{2} \)
97 \( 1 + (1.26e3 - 727. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.30066257468982239679412342948, −10.41011809265322427822695133035, −9.827388423560296815932762272808, −8.660751285475670204970080970681, −7.20039353253786108475492258100, −6.46542215830221123835601263683, −5.15720389165712947088965895228, −3.79876272472512487332769382975, −3.02590906922879015719005499403, −0.21553202770322213509818458046, 1.19397318796447885983325835127, 2.95342330080750196778633930197, 4.62158786790280445939948216028, 5.77292355519703674322074429631, 6.72675741254342067842480196776, 7.88135928351014699324747172201, 8.684403661503451397099225008452, 9.798765896126517224453913200525, 11.04323826262343928164618143110, 12.20519560331921129329411193265

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