Properties

Label 2-252-63.41-c3-0-6
Degree $2$
Conductor $252$
Sign $-0.984 - 0.175i$
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  + (3.66 + 3.68i)3-s + (−8.29 + 14.3i)5-s + (−6.28 + 17.4i)7-s + (−0.127 + 26.9i)9-s + (46.2 − 26.7i)11-s + (−11.0 − 6.37i)13-s + (−83.3 + 22.1i)15-s − 96.7·17-s − 54.6i·19-s + (−87.1 + 40.7i)21-s + (−55.6 − 32.1i)23-s + (−75.2 − 130. i)25-s + (−99.9 + 98.5i)27-s + (−112. + 65.0i)29-s + (190. + 110. i)31-s + ⋯
L(s)  = 1  + (0.705 + 0.708i)3-s + (−0.742 + 1.28i)5-s + (−0.339 + 0.940i)7-s + (−0.00470 + 0.999i)9-s + (1.26 − 0.732i)11-s + (−0.235 − 0.136i)13-s + (−1.43 + 0.380i)15-s − 1.38·17-s − 0.660i·19-s + (−0.906 + 0.423i)21-s + (−0.504 − 0.291i)23-s + (−0.601 − 1.04i)25-s + (−0.712 + 0.702i)27-s + (−0.721 + 0.416i)29-s + (1.10 + 0.637i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.984 - 0.175i$
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: \((2,\ 252,\ (\ :3/2),\ -0.984 - 0.175i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.424432837\)
\(L(\frac12)\) \(\approx\) \(1.424432837\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.66 - 3.68i)T \)
7 \( 1 + (6.28 - 17.4i)T \)
good5 \( 1 + (8.29 - 14.3i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-46.2 + 26.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (11.0 + 6.37i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 96.7T + 4.91e3T^{2} \)
19 \( 1 + 54.6iT - 6.85e3T^{2} \)
23 \( 1 + (55.6 + 32.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (112. - 65.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-190. - 110. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 279.T + 5.06e4T^{2} \)
41 \( 1 + (185. - 320. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (153. + 266. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-163. - 283. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 451. iT - 1.48e5T^{2} \)
59 \( 1 + (-258. + 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-234. + 135. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (370. - 642. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 914. iT - 3.57e5T^{2} \)
73 \( 1 - 337. iT - 3.89e5T^{2} \)
79 \( 1 + (498. + 863. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-17.0 - 29.4i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 208.T + 7.04e5T^{2} \)
97 \( 1 + (-1.10e3 + 635. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.63579123577736223336104223357, −11.22134975055910173670610201787, −10.11415271064344491811588385133, −9.067368277200032756319144836875, −8.393379703790211331248314074882, −7.06728540534389974494801607582, −6.15905076696325018378537788008, −4.46677431190655788317040816577, −3.36826403145097574039839785676, −2.50400414996703955298423821527, 0.51206479762816035786765542357, 1.80110162713730102301654237897, 3.84230974143311907867352254755, 4.42633682828587348205565827597, 6.36451737248830121084696491669, 7.30100463601776284267300032227, 8.185531177910482329894512744646, 9.095099687311501401004473957434, 9.861361673195024553845920413053, 11.56273586736142487873998904340

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