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Properties

Label 2-252-63.25-c3-0-8
Degree $2$
Conductor $252$
Sign $0.891 - 0.452i$
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  + (−4.33 + 2.86i)3-s + (0.863 − 1.49i)5-s + (−16.5 − 8.25i)7-s + (10.5 − 24.8i)9-s + (28.7 + 49.7i)11-s + (−31.7 − 54.9i)13-s + (0.547 + 8.95i)15-s + (27.1 − 47.0i)17-s + (30.6 + 53.0i)19-s + (95.5 − 11.7i)21-s + (41.0 − 71.1i)23-s + (61.0 + 105. i)25-s + (25.5 + 137. i)27-s + (−97.6 + 169. i)29-s + 305.·31-s + ⋯
L(s)  = 1  + (−0.833 + 0.551i)3-s + (0.0772 − 0.133i)5-s + (−0.895 − 0.445i)7-s + (0.390 − 0.920i)9-s + (0.788 + 1.36i)11-s + (−0.677 − 1.17i)13-s + (0.00941 + 0.154i)15-s + (0.387 − 0.671i)17-s + (0.370 + 0.640i)19-s + (0.992 − 0.122i)21-s + (0.372 − 0.644i)23-s + (0.488 + 0.845i)25-s + (0.181 + 0.983i)27-s + (−0.625 + 1.08i)29-s + 1.77·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.891 - 0.452i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.891 - 0.452i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.188512492\)
\(L(\frac12)\) \(\approx\) \(1.188512492\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.33 - 2.86i)T \)
7 \( 1 + (16.5 + 8.25i)T \)
good5 \( 1 + (-0.863 + 1.49i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.7 - 49.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (31.7 + 54.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-27.1 + 47.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-30.6 - 53.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-41.0 + 71.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (97.6 - 169. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 305.T + 2.97e4T^{2} \)
37 \( 1 + (-125. - 217. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (23.5 + 40.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (48.6 - 84.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 635.T + 1.03e5T^{2} \)
53 \( 1 + (-304. + 527. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 372.T + 2.05e5T^{2} \)
61 \( 1 - 187.T + 2.26e5T^{2} \)
67 \( 1 + 1.01e3T + 3.00e5T^{2} \)
71 \( 1 + 772.T + 3.57e5T^{2} \)
73 \( 1 + (-139. + 240. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + (-140. + 242. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (368. + 638. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-167. + 290. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.84692001944661207744634805028, −10.41917243714716738005256220689, −9.972733617272158072944348754487, −9.147501100185571228496963411669, −7.42206513773388231558053728096, −6.66405683995707283354655477290, −5.42206672636458658012121338387, −4.45085422461543424125568116557, −3.15185380199200478563557576620, −0.885940821935140688211858126500, 0.78331809588258396060234888579, 2.56289450486287381760760469177, 4.17474364282719145384644427175, 5.73453209794287271532636263378, 6.35399463269676180139710252289, 7.29607202373255664476508989646, 8.685791959802386064592851479502, 9.624462937576429446157387228560, 10.74075748959492861545055533100, 11.76753184844068045529013909588

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