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Properties

Label 2-252-12.11-c3-0-6
Degree $2$
Conductor $252$
Sign $-0.902 + 0.429i$
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  + (1.42 + 2.44i)2-s + (−3.91 + 6.97i)4-s + 15.7i·5-s + 7i·7-s + (−22.6 + 0.420i)8-s + (−38.5 + 22.5i)10-s + 23.5·11-s − 37.7·13-s + (−17.0 + 10.0i)14-s + (−33.3 − 54.6i)16-s + 31.9i·17-s − 28.8i·19-s + (−110. − 61.7i)20-s + (33.6 + 57.3i)22-s + 50.8·23-s + ⋯
L(s)  = 1  + (0.505 + 0.862i)2-s + (−0.489 + 0.872i)4-s + 1.41i·5-s + 0.377i·7-s + (−0.999 + 0.0186i)8-s + (−1.21 + 0.713i)10-s + 0.644·11-s − 0.804·13-s + (−0.326 + 0.191i)14-s + (−0.521 − 0.853i)16-s + 0.455i·17-s − 0.348i·19-s + (−1.23 − 0.690i)20-s + (0.325 + 0.556i)22-s + 0.461·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.902 + 0.429i$
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: \((2,\ 252,\ (\ :3/2),\ -0.902 + 0.429i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.590464430\)
\(L(\frac12)\) \(\approx\) \(1.590464430\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.42 - 2.44i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 - 15.7iT - 125T^{2} \)
11 \( 1 - 23.5T + 1.33e3T^{2} \)
13 \( 1 + 37.7T + 2.19e3T^{2} \)
17 \( 1 - 31.9iT - 4.91e3T^{2} \)
19 \( 1 + 28.8iT - 6.85e3T^{2} \)
23 \( 1 - 50.8T + 1.21e4T^{2} \)
29 \( 1 + 205. iT - 2.43e4T^{2} \)
31 \( 1 - 176. iT - 2.97e4T^{2} \)
37 \( 1 + 338.T + 5.06e4T^{2} \)
41 \( 1 + 77.7iT - 6.89e4T^{2} \)
43 \( 1 - 463. iT - 7.95e4T^{2} \)
47 \( 1 + 352.T + 1.03e5T^{2} \)
53 \( 1 - 155. iT - 1.48e5T^{2} \)
59 \( 1 - 761.T + 2.05e5T^{2} \)
61 \( 1 + 40.2T + 2.26e5T^{2} \)
67 \( 1 - 838. iT - 3.00e5T^{2} \)
71 \( 1 + 70.1T + 3.57e5T^{2} \)
73 \( 1 - 726.T + 3.89e5T^{2} \)
79 \( 1 + 992. iT - 4.93e5T^{2} \)
83 \( 1 - 1.11e3T + 5.71e5T^{2} \)
89 \( 1 - 1.59e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.26e3T + 9.12e5T^{2} \)
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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

−12.16955070328228762207391558016, −11.38421375244105838601736311819, −10.19232972670068643680616876324, −9.127491385800255173402369046346, −7.955416504372472222662032156891, −6.92364435456622317253145370472, −6.36025562757182506790424055289, −5.07654253718620124510930467448, −3.69086699158161750601291868816, −2.58997822615342027481774248801, 0.53091047434298585754218325559, 1.79207074073296743170844719269, 3.56994140427078625172883513622, 4.71287497674226534094068884572, 5.40828259499931867448741115448, 6.93276936615169854797719413930, 8.474696701403061302008318166238, 9.281128784609251987758180404201, 10.08552558799384951898678767575, 11.27694379610669485328830229721

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