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Properties

Label 2-252-63.25-c1-0-4
Degree $2$
Conductor $252$
Sign $-0.0124 + 0.999i$
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  + (−1.29 − 1.15i)3-s + (0.381 − 0.661i)5-s + (2.62 − 0.297i)7-s + (0.337 + 2.98i)9-s + (−3.01 − 5.22i)11-s + (−1.26 − 2.18i)13-s + (−1.25 + 0.413i)15-s + (1.94 − 3.36i)17-s + (−2.13 − 3.69i)19-s + (−3.73 − 2.64i)21-s + (0.732 − 1.26i)23-s + (2.20 + 3.82i)25-s + (3.00 − 4.23i)27-s + (−3.00 + 5.20i)29-s + 6.56·31-s + ⋯
L(s)  = 1  + (−0.745 − 0.666i)3-s + (0.170 − 0.295i)5-s + (0.993 − 0.112i)7-s + (0.112 + 0.993i)9-s + (−0.909 − 1.57i)11-s + (−0.349 − 0.605i)13-s + (−0.324 + 0.106i)15-s + (0.471 − 0.816i)17-s + (−0.489 − 0.848i)19-s + (−0.816 − 0.577i)21-s + (0.152 − 0.264i)23-s + (0.441 + 0.764i)25-s + (0.578 − 0.815i)27-s + (−0.558 + 0.967i)29-s + 1.17·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0124 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0124 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0124 + 0.999i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.0124 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701807 - 0.710575i\)
\(L(\frac12)\) \(\approx\) \(0.701807 - 0.710575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.29 + 1.15i)T \)
7 \( 1 + (-2.62 + 0.297i)T \)
good5 \( 1 + (-0.381 + 0.661i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.01 + 5.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.26 + 2.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.94 + 3.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.13 + 3.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.732 + 1.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.00 - 5.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + (-4.82 - 8.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.24 + 3.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.13 - 3.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.77T + 47T^{2} \)
53 \( 1 + (0.265 - 0.459i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 8.38T + 61T^{2} \)
67 \( 1 + 1.92T + 67T^{2} \)
71 \( 1 - 9.90T + 71T^{2} \)
73 \( 1 + (2.13 - 3.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.40T + 79T^{2} \)
83 \( 1 + (8.05 - 13.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.76 - 3.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.33 + 4.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.56913760402192511472704638394, −11.13395055156887335498751068219, −10.16989509497229664504442746326, −8.569257823666969296272162804795, −7.924557159779651460675015551448, −6.79584595499336089933153858161, −5.44981143881997659850216974681, −4.96276450069888207763441442698, −2.78680738046876113655481131169, −0.916857380986240354336948022301, 2.09294840209048169578169593544, 4.15376564242794447626869501175, 4.96513759148115413019164046147, 6.07494840641167477845081502189, 7.30781278167085495310746597071, 8.370482366829702615989816036296, 9.857887614316872380591493805684, 10.23993139458219714190402709116, 11.31246006485294572121613115472, 12.12770810251136992755584522935

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