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Properties

Label 2-252-28.27-c3-0-32
Degree $2$
Conductor $252$
Sign $1$
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  + 2.82i·2-s − 8.00·4-s − 3.74i·5-s − 18.5·7-s − 22.6i·8-s + 10.5·10-s + 43.8i·11-s − 52.3i·14-s + 64.0·16-s − 138. i·17-s + 153.·19-s + 29.9i·20-s − 124·22-s − 134. i·23-s + 111·25-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 0.334i·5-s − 0.999·7-s − 1.00i·8-s + 0.334·10-s + 1.20i·11-s − 1.00i·14-s + 1.00·16-s − 1.97i·17-s + 1.85·19-s + 0.334i·20-s − 1.20·22-s − 1.21i·23-s + 0.888·25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $1$
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: \((2,\ 252,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.200238227\)
\(L(\frac12)\) \(\approx\) \(1.200238227\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 \)
7 \( 1 + 18.5T \)
good5 \( 1 + 3.74iT - 125T^{2} \)
11 \( 1 - 43.8iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 138. iT - 4.91e3T^{2} \)
19 \( 1 - 153.T + 6.85e3T^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 201.T + 2.97e4T^{2} \)
37 \( 1 + 376T + 5.06e4T^{2} \)
41 \( 1 + 407. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 - 451. iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 1.55e3iT - 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−11.99127259392016934486902911651, −10.17834688623059297945152922930, −9.541677583556514185751943801993, −8.731676125963629272003867324043, −7.26200532661797214383933458606, −6.89711445878708802742566538418, −5.42034267071324018694037001188, −4.59283337708448629931303606356, −3.04047188582920982902370406305, −0.57690349490050162402160031150, 1.17917962662012441699196501148, 3.05239868662989753285474638581, 3.65146401890676786754895521764, 5.37217377469908732004529788107, 6.38827593642248416515046915610, 7.929308167024488456621050903713, 8.929272523648213657811712622884, 9.892064803671367272456262801476, 10.64125563789830382902156871542, 11.56306670143881291963707247527

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