Properties

Label 2-252-28.27-c3-0-2
Degree $2$
Conductor $252$
Sign $-0.267 + 0.963i$
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  + (0.951 + 2.66i)2-s + (−6.19 + 5.06i)4-s + 10.8i·5-s + (−15.1 + 10.6i)7-s + (−19.3 − 11.6i)8-s + (−28.8 + 10.3i)10-s − 45.0i·11-s + 40.3i·13-s + (−42.8 − 30.1i)14-s + (12.6 − 62.7i)16-s − 8.11i·17-s − 53.3·19-s + (−54.9 − 67.1i)20-s + (119. − 42.8i)22-s − 55.7i·23-s + ⋯
L(s)  = 1  + (0.336 + 0.941i)2-s + (−0.773 + 0.633i)4-s + 0.970i·5-s + (−0.817 + 0.576i)7-s + (−0.856 − 0.515i)8-s + (−0.913 + 0.326i)10-s − 1.23i·11-s + 0.861i·13-s + (−0.817 − 0.575i)14-s + (0.197 − 0.980i)16-s − 0.115i·17-s − 0.643·19-s + (−0.614 − 0.750i)20-s + (1.16 − 0.415i)22-s − 0.505i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.267 + 0.963i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.267 + 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4885942994\)
\(L(\frac12)\) \(\approx\) \(0.4885942994\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 2.66i)T \)
3 \( 1 \)
7 \( 1 + (15.1 - 10.6i)T \)
good5 \( 1 - 10.8iT - 125T^{2} \)
11 \( 1 + 45.0iT - 1.33e3T^{2} \)
13 \( 1 - 40.3iT - 2.19e3T^{2} \)
17 \( 1 + 8.11iT - 4.91e3T^{2} \)
19 \( 1 + 53.3T + 6.85e3T^{2} \)
23 \( 1 + 55.7iT - 1.21e4T^{2} \)
29 \( 1 - 169.T + 2.43e4T^{2} \)
31 \( 1 + 262.T + 2.97e4T^{2} \)
37 \( 1 + 354.T + 5.06e4T^{2} \)
41 \( 1 + 42.7iT - 6.89e4T^{2} \)
43 \( 1 + 23.1iT - 7.95e4T^{2} \)
47 \( 1 + 437.T + 1.03e5T^{2} \)
53 \( 1 + 388.T + 1.48e5T^{2} \)
59 \( 1 - 649.T + 2.05e5T^{2} \)
61 \( 1 - 916. iT - 2.26e5T^{2} \)
67 \( 1 + 736. iT - 3.00e5T^{2} \)
71 \( 1 - 600. iT - 3.57e5T^{2} \)
73 \( 1 - 673. iT - 3.89e5T^{2} \)
79 \( 1 - 585. iT - 4.93e5T^{2} \)
83 \( 1 + 856.T + 5.71e5T^{2} \)
89 \( 1 + 920. iT - 7.04e5T^{2} \)
97 \( 1 - 638. iT - 9.12e5T^{2} \)
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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

−12.44561343758247341714748811353, −11.41350094077903719463479204815, −10.30093616222840185098582049453, −9.090089411634967730803950817398, −8.397390272238430774576338389765, −6.91269084318761826726684641760, −6.46279500200137004810695115178, −5.39806985034113767291618172465, −3.80825539119958056317649513540, −2.81131174362307648565906204323, 0.17013813352285516130640906202, 1.65257225838441165855640533742, 3.29955416054073761995044520666, 4.46226540784416266957681415399, 5.37444573187218228748386099702, 6.79162098659087742881528389988, 8.226405649666813029758281426857, 9.306597305466872540150590391814, 10.03359205256711011727937862757, 10.84677757955111882754615140536

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