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Properties

Label 2-252-12.11-c3-0-34
Degree $2$
Conductor $252$
Sign $-0.666 + 0.745i$
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  + (1.88 + 2.11i)2-s + (−0.909 + 7.94i)4-s − 21.8i·5-s + 7i·7-s + (−18.4 + 13.0i)8-s + (46.1 − 41.1i)10-s − 66.1·11-s − 61.5·13-s + (−14.7 + 13.1i)14-s + (−62.3 − 14.4i)16-s − 11.7i·17-s + 87.9i·19-s + (173. + 19.8i)20-s + (−124. − 139. i)22-s + 13.2·23-s + ⋯
L(s)  = 1  + (0.665 + 0.746i)2-s + (−0.113 + 0.993i)4-s − 1.95i·5-s + 0.377i·7-s + (−0.817 + 0.576i)8-s + (1.45 − 1.30i)10-s − 1.81·11-s − 1.31·13-s + (−0.282 + 0.251i)14-s + (−0.974 − 0.225i)16-s − 0.167i·17-s + 1.06i·19-s + (1.94 + 0.222i)20-s + (−1.20 − 1.35i)22-s + 0.119·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.666 + 0.745i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3404608460\)
\(L(\frac12)\) \(\approx\) \(0.3404608460\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.88 - 2.11i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 + 21.8iT - 125T^{2} \)
11 \( 1 + 66.1T + 1.33e3T^{2} \)
13 \( 1 + 61.5T + 2.19e3T^{2} \)
17 \( 1 + 11.7iT - 4.91e3T^{2} \)
19 \( 1 - 87.9iT - 6.85e3T^{2} \)
23 \( 1 - 13.2T + 1.21e4T^{2} \)
29 \( 1 - 7.53iT - 2.43e4T^{2} \)
31 \( 1 + 45.1iT - 2.97e4T^{2} \)
37 \( 1 - 207.T + 5.06e4T^{2} \)
41 \( 1 + 172. iT - 6.89e4T^{2} \)
43 \( 1 + 331. iT - 7.95e4T^{2} \)
47 \( 1 - 436.T + 1.03e5T^{2} \)
53 \( 1 + 282. iT - 1.48e5T^{2} \)
59 \( 1 - 553.T + 2.05e5T^{2} \)
61 \( 1 + 262.T + 2.26e5T^{2} \)
67 \( 1 + 89.8iT - 3.00e5T^{2} \)
71 \( 1 + 891.T + 3.57e5T^{2} \)
73 \( 1 + 506.T + 3.89e5T^{2} \)
79 \( 1 - 836. iT - 4.93e5T^{2} \)
83 \( 1 + 114.T + 5.71e5T^{2} \)
89 \( 1 + 162. iT - 7.04e5T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−11.81459841283539351300633986002, −10.07777230356818397390368593218, −9.015291516746877123049740864654, −8.139184812939871322125595882609, −7.49075484420588953575666994591, −5.59519442804877853441623718517, −5.24431923516112754360711505211, −4.23359262839044893939915801818, −2.39380043906675767039569567797, −0.097213382828791341623580477369, 2.52063851231095441804275411816, 2.92459240640868214941759809736, 4.51070108107426160528074439583, 5.75182809248377210820779995854, 6.93710035470479005338524637818, 7.67884293894270456633856474014, 9.640247144248555948136589237036, 10.41207204312902088108037994370, 10.86919758774321137399942166954, 11.76449443404940317943011568741

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