Properties

Label 2-252-252.139-c1-0-10
Degree $2$
Conductor $252$
Sign $0.934 - 0.356i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)2-s − 1.73·3-s + (−1.73 − i)4-s + (−1.5 − 0.866i)5-s + (0.633 − 2.36i)6-s + (2.59 − 0.5i)7-s + (2 − 1.99i)8-s + 2.99·9-s + (1.73 − 1.73i)10-s + (−0.866 + 0.5i)11-s + (2.99 + 1.73i)12-s + (1.5 + 0.866i)13-s + (−0.267 + 3.73i)14-s + (2.59 + 1.49i)15-s + (1.99 + 3.46i)16-s − 3.46i·17-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s − 1.00·3-s + (−0.866 − 0.5i)4-s + (−0.670 − 0.387i)5-s + (0.258 − 0.965i)6-s + (0.981 − 0.188i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s + (0.547 − 0.547i)10-s + (−0.261 + 0.150i)11-s + (0.866 + 0.499i)12-s + (0.416 + 0.240i)13-s + (−0.0716 + 0.997i)14-s + (0.670 + 0.387i)15-s + (0.499 + 0.866i)16-s − 0.840i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.728137 + 0.134251i\)
\(L(\frac12)\) \(\approx\) \(0.728137 + 0.134251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 + 1.73T \)
7 \( 1 + (-2.59 + 0.5i)T \)
good5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + (-4.33 - 2.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.33 + 7.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (-7.5 - 4.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.9 + 7.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + (-2.59 + 1.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.866 - 1.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 17.3iT - 89T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.80645776447348793380029164009, −11.34848341954288387488874756433, −10.09959275969914069405439866245, −9.124095635779032269498349203989, −7.76125256263656165143499350037, −7.38877656479686805733736527191, −5.95186984176749184658675205634, −5.00185437173545080128885535718, −4.21545744579474469985517269000, −0.938775135111532964902072001335, 1.29992899800338632119095107146, 3.30143273187698446438651613300, 4.60679245217432409351257916386, 5.55649059205630938416890530759, 7.24939555692379357938558626047, 8.099691198016922819649457921691, 9.277888819617372154354207277328, 10.70033374824078062119403571241, 10.89651310590881379391728429451, 11.83666845495690266113330351057

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