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Properties

Label 2-252-63.20-c3-0-6
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: $\chi_{252} (209, \cdot )$
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

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

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