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Properties

Label 2-252-63.5-c3-0-17
Degree $2$
Conductor $252$
Sign $-0.251 + 0.967i$
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  + (0.973 + 5.10i)3-s + (−6.29 − 10.8i)5-s + (18.2 + 2.87i)7-s + (−25.1 + 9.94i)9-s + (−59.5 − 34.3i)11-s + (34.1 + 19.7i)13-s + (49.4 − 42.7i)15-s + (−38.3 − 66.4i)17-s + (−69.3 − 40.0i)19-s + (3.13 + 96.1i)21-s + (−134. + 77.7i)23-s + (−16.6 + 28.8i)25-s + (−75.1 − 118. i)27-s + (197. − 114. i)29-s − 145. i·31-s + ⋯
L(s)  = 1  + (0.187 + 0.982i)3-s + (−0.562 − 0.974i)5-s + (0.987 + 0.155i)7-s + (−0.929 + 0.368i)9-s + (−1.63 − 0.942i)11-s + (0.728 + 0.420i)13-s + (0.851 − 0.735i)15-s + (−0.547 − 0.948i)17-s + (−0.836 − 0.483i)19-s + (0.0325 + 0.999i)21-s + (−1.22 + 0.704i)23-s + (−0.133 + 0.230i)25-s + (−0.535 − 0.844i)27-s + (1.26 − 0.731i)29-s − 0.842i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.251 + 0.967i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.251 + 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8102206781\)
\(L(\frac12)\) \(\approx\) \(0.8102206781\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.973 - 5.10i)T \)
7 \( 1 + (-18.2 - 2.87i)T \)
good5 \( 1 + (6.29 + 10.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (59.5 + 34.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-34.1 - 19.7i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (38.3 + 66.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.3 + 40.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (134. - 77.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-197. + 114. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 145. iT - 2.97e4T^{2} \)
37 \( 1 + (-74.1 + 128. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-27.9 + 48.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (109. + 188. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 109.T + 1.03e5T^{2} \)
53 \( 1 + (257. - 148. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 179.T + 2.05e5T^{2} \)
61 \( 1 + 318. iT - 2.26e5T^{2} \)
67 \( 1 + 653.T + 3.00e5T^{2} \)
71 \( 1 + 157. iT - 3.57e5T^{2} \)
73 \( 1 + (1.05e3 - 609. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 818.T + 4.93e5T^{2} \)
83 \( 1 + (-404. - 700. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-20.9 + 36.3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (161. - 93.1i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.22860202676477771647976653202, −10.53627162605526401827240174410, −9.206785615035961538467365255681, −8.367962929968332189577490552027, −7.919237144688902353060670348414, −5.84086064110298615904286029233, −4.88155407624439200031594172995, −4.13216475028248775959335289437, −2.50435388563123359396558559549, −0.29829881044114212446747180079, 1.78364785245363876452565648161, 2.96307690765775680791972928259, 4.54044887330585897449723816633, 6.03744698621982943953378708862, 7.05493513382860803599462873731, 8.038150304364127972251554285305, 8.365729011367091166222394099827, 10.50255452972634778348758258414, 10.72429317702264163999668674492, 11.97828919989741399561829162060

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