Properties

Label 2-252-63.41-c3-0-20
Degree $2$
Conductor $252$
Sign $-0.106 + 0.994i$
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  + (4.66 − 2.29i)3-s + (5.16 − 8.94i)5-s + (−17.2 + 6.79i)7-s + (16.4 − 21.3i)9-s + (27.1 − 15.6i)11-s + (−39.0 − 22.5i)13-s + (3.57 − 53.5i)15-s + 62.4·17-s − 132. i·19-s + (−64.7 + 71.1i)21-s + (−58.8 − 33.9i)23-s + (9.14 + 15.8i)25-s + (27.8 − 137. i)27-s + (−116. + 67.3i)29-s + (−25.9 − 15.0i)31-s + ⋯
L(s)  = 1  + (0.897 − 0.441i)3-s + (0.461 − 0.800i)5-s + (−0.930 + 0.366i)7-s + (0.610 − 0.791i)9-s + (0.743 − 0.429i)11-s + (−0.832 − 0.480i)13-s + (0.0616 − 0.921i)15-s + 0.891·17-s − 1.60i·19-s + (−0.673 + 0.739i)21-s + (−0.533 − 0.307i)23-s + (0.0731 + 0.126i)25-s + (0.198 − 0.980i)27-s + (−0.747 + 0.431i)29-s + (−0.150 − 0.0869i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.106 + 0.994i$
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: \((2,\ 252,\ (\ :3/2),\ -0.106 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.270995685\)
\(L(\frac12)\) \(\approx\) \(2.270995685\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.66 + 2.29i)T \)
7 \( 1 + (17.2 - 6.79i)T \)
good5 \( 1 + (-5.16 + 8.94i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-27.1 + 15.6i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (39.0 + 22.5i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 62.4T + 4.91e3T^{2} \)
19 \( 1 + 132. iT - 6.85e3T^{2} \)
23 \( 1 + (58.8 + 33.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (116. - 67.3i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (25.9 + 15.0i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 40.4T + 5.06e4T^{2} \)
41 \( 1 + (-39.7 + 68.8i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-161. - 279. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (171. + 296. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 64.9iT - 1.48e5T^{2} \)
59 \( 1 + (-79.3 + 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-493. + 285. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-150. + 261. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 719. iT - 3.57e5T^{2} \)
73 \( 1 - 558. iT - 3.89e5T^{2} \)
79 \( 1 + (-456. - 790. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (352. + 610. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 700.T + 7.04e5T^{2} \)
97 \( 1 + (202. - 117. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.58578455994361498857081715028, −9.953198574190013979526453716299, −9.308595496197501599282940980867, −8.640971269971374211714755495367, −7.41306878892125456385374231832, −6.39005271487791147981944466479, −5.17587594736026128612389131386, −3.59621638408992711248424668133, −2.43045710614774818576793938780, −0.816551139570443416644424288248, 1.95522974336015084239781083114, 3.25797483459201228646082910708, 4.16943253656688757460740896713, 5.87954552003167504874137637777, 7.00157640881533514532914423881, 7.83847881140922913519413775975, 9.303551502314464907775006310386, 9.892143768886041055048814687046, 10.45450430214371363697306885626, 11.94969661668236637403217715399

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