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Properties

Label 2-252-252.139-c1-0-9
Degree $2$
Conductor $252$
Sign $0.984 - 0.177i$
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  + (−1.13 − 0.842i)2-s + (1.59 + 0.677i)3-s + (0.581 + 1.91i)4-s + (−1.43 − 0.827i)5-s + (−1.24 − 2.11i)6-s + (−0.0302 + 2.64i)7-s + (0.950 − 2.66i)8-s + (2.08 + 2.15i)9-s + (0.931 + 2.14i)10-s + (4.24 − 2.45i)11-s + (−0.368 + 3.44i)12-s + (0.876 + 0.505i)13-s + (2.26 − 2.98i)14-s + (−1.72 − 2.28i)15-s + (−3.32 + 2.22i)16-s + 7.66i·17-s + ⋯
L(s)  = 1  + (−0.803 − 0.595i)2-s + (0.920 + 0.391i)3-s + (0.290 + 0.956i)4-s + (−0.640 − 0.369i)5-s + (−0.506 − 0.862i)6-s + (−0.0114 + 0.999i)7-s + (0.336 − 0.941i)8-s + (0.694 + 0.719i)9-s + (0.294 + 0.678i)10-s + (1.28 − 0.739i)11-s + (−0.106 + 0.994i)12-s + (0.243 + 0.140i)13-s + (0.604 − 0.796i)14-s + (−0.444 − 0.590i)15-s + (−0.830 + 0.556i)16-s + 1.85i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.984 - 0.177i$
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: \((2,\ 252,\ (\ :1/2),\ 0.984 - 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11440 + 0.0996269i\)
\(L(\frac12)\) \(\approx\) \(1.11440 + 0.0996269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.842i)T \)
3 \( 1 + (-1.59 - 0.677i)T \)
7 \( 1 + (0.0302 - 2.64i)T \)
good5 \( 1 + (1.43 + 0.827i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.24 + 2.45i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.876 - 0.505i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.66iT - 17T^{2} \)
19 \( 1 - 2.85T + 19T^{2} \)
23 \( 1 + (-1.98 - 1.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.19 + 7.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.348 - 0.604i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.97T + 37T^{2} \)
41 \( 1 + (6.69 + 3.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.70 + 3.87i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.70 + 6.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.61T + 53T^{2} \)
59 \( 1 + (2.54 - 4.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.12 - 2.96i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.18 + 1.26i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.00iT - 71T^{2} \)
73 \( 1 + 5.89iT - 73T^{2} \)
79 \( 1 + (4.90 - 2.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.20 - 2.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.19iT - 89T^{2} \)
97 \( 1 + (-8.82 + 5.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.90465965177883323983104492340, −11.14124296545307637244301834209, −9.944874440513360164320212425793, −8.974616818508500090900440982747, −8.553138899287480889767511709518, −7.68426975041990872307014949602, −6.14277077865000502846535756291, −4.16385498181209227796423907536, −3.37043204859721118295726424332, −1.77780045336498452885415942401, 1.27641179419248959828510055389, 3.24605204700666122093799458481, 4.64937239645300082633294476265, 6.63968927975448069014990157259, 7.26044220022708919617363684120, 7.82719980455643057604434872504, 9.250815659754642281998169747138, 9.602803708148709845495916548757, 10.98467286583371351562631488651, 11.79695635931859023960800931784

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