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Properties

Label 2-252-63.38-c3-0-0
Degree $2$
Conductor $252$
Sign $-0.973 + 0.230i$
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  + (−0.452 + 5.17i)3-s + (6.54 − 11.3i)5-s + (1.26 − 18.4i)7-s + (−26.5 − 4.68i)9-s + (−45.6 + 26.3i)11-s + (−31.8 + 18.3i)13-s + (55.7 + 39.0i)15-s + (−20.4 + 35.4i)17-s + (−108. + 62.8i)19-s + (95.0 + 14.9i)21-s + (1.99 + 1.14i)23-s + (−23.1 − 40.0i)25-s + (36.3 − 135. i)27-s + (−151. − 87.5i)29-s + 268. i·31-s + ⋯
L(s)  = 1  + (−0.0871 + 0.996i)3-s + (0.585 − 1.01i)5-s + (0.0685 − 0.997i)7-s + (−0.984 − 0.173i)9-s + (−1.25 + 0.722i)11-s + (−0.679 + 0.392i)13-s + (0.958 + 0.671i)15-s + (−0.292 + 0.506i)17-s + (−1.31 + 0.758i)19-s + (0.987 + 0.155i)21-s + (0.0180 + 0.0104i)23-s + (−0.185 − 0.320i)25-s + (0.258 − 0.965i)27-s + (−0.971 − 0.560i)29-s + 1.55i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.973 + 0.230i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.973 + 0.230i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.02741197915\)
\(L(\frac12)\) \(\approx\) \(0.02741197915\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.452 - 5.17i)T \)
7 \( 1 + (-1.26 + 18.4i)T \)
good5 \( 1 + (-6.54 + 11.3i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (45.6 - 26.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (31.8 - 18.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (20.4 - 35.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (108. - 62.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-1.99 - 1.14i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (151. + 87.5i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 268. iT - 2.97e4T^{2} \)
37 \( 1 + (191. + 331. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (75.5 + 130. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-23.2 + 40.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 372.T + 1.03e5T^{2} \)
53 \( 1 + (-539. - 311. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 4.01T + 2.05e5T^{2} \)
61 \( 1 + 779. iT - 2.26e5T^{2} \)
67 \( 1 + 439.T + 3.00e5T^{2} \)
71 \( 1 - 800. iT - 3.57e5T^{2} \)
73 \( 1 + (428. + 247. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 140.T + 4.93e5T^{2} \)
83 \( 1 + (-506. + 877. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-211. - 366. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (264. + 152. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.25778590519389704316212744435, −10.67614602727384357119441858383, −10.36232026311370480949058694786, −9.330231184371446114853824395170, −8.448161901745808939855531963887, −7.23774081314026179847187889174, −5.68644195436119016399809687811, −4.81081481060909512541324722499, −3.95742660677340239554018482206, −2.04840004948245426138634126182, 0.009785305991716759405084855363, 2.31684873136777361714918505188, 2.79055867596661221316884143080, 5.23381508102808358086525033671, 6.05754852136654496641300118936, 6.99937725240066135304310495585, 8.042157476664408843002275548795, 8.993112582564228074402794416188, 10.34246367292800615651932848310, 11.12556620374719359883724372608

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