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Properties

Label 2-252-252.223-c1-0-25
Degree $2$
Conductor $252$
Sign $0.513 + 0.858i$
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 + (1.73 + 2i)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 + (2.09 − 3.09i)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 + (0.654 + 0.755i)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 + (0.560 − 0.827i)14-s + (0.670 − 0.387i)15-s + (0.499 − 0.866i)16-s − 0.840i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.513 + 0.858i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.513 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37022 - 0.776922i\)
\(L(\frac12)\) \(\approx\) \(1.37022 - 0.776922i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 2i)T \)
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

−11.91256087905252943417691808881, −10.86319802101227006211718321875, −9.824459961999022875327573469240, −8.976855697776746668278763834093, −8.506660253201418285764253109204, −7.27993512837188473218176482476, −5.37282048949243517978431143646, −4.31715631829141261381732144300, −2.71876645157251127892315917193, −1.81910589390400723078534820896, 1.89256466737531534176309879737, 3.84598860026730180288414799812, 5.00909661241390223590312293124, 6.45462138982227393152421919620, 7.36551539489048063124601893886, 8.214232252471624490554566547961, 9.080384023916151986944340583171, 10.22246575129054715398872960475, 10.64946325962713129202321370679, 12.69769979872109965701836316379

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