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Properties

Label 2-252-63.16-c1-0-3
Degree $2$
Conductor $252$
Sign $0.943 - 0.332i$
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  + (1.68 + 0.403i)3-s + 3.60·5-s + (−1.60 + 2.10i)7-s + (2.67 + 1.35i)9-s − 6.02·11-s + (−2.55 − 4.42i)13-s + (6.06 + 1.45i)15-s + (0.111 + 0.192i)17-s + (1.71 − 2.96i)19-s + (−3.54 + 2.89i)21-s − 1.01·23-s + 7.98·25-s + (3.95 + 3.36i)27-s + (−2.83 + 4.91i)29-s + (2.52 − 4.37i)31-s + ⋯
L(s)  = 1  + (0.972 + 0.232i)3-s + 1.61·5-s + (−0.605 + 0.795i)7-s + (0.891 + 0.452i)9-s − 1.81·11-s + (−0.709 − 1.22i)13-s + (1.56 + 0.375i)15-s + (0.0269 + 0.0466i)17-s + (0.392 − 0.680i)19-s + (−0.774 + 0.632i)21-s − 0.212·23-s + 1.59·25-s + (0.761 + 0.648i)27-s + (−0.526 + 0.912i)29-s + (0.453 − 0.784i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.943 - 0.332i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82983 + 0.312821i\)
\(L(\frac12)\) \(\approx\) \(1.82983 + 0.312821i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.403i)T \)
7 \( 1 + (1.60 - 2.10i)T \)
good5 \( 1 - 3.60T + 5T^{2} \)
11 \( 1 + 6.02T + 11T^{2} \)
13 \( 1 + (2.55 + 4.42i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.111 - 0.192i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.71 + 2.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 + (2.83 - 4.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.52 + 4.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.68 + 2.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0955 - 0.165i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.71 + 2.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.03 - 1.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.65 + 4.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.891 + 1.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.49 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.89T + 71T^{2} \)
73 \( 1 + (-6.30 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.30 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.59 - 6.23i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.44 - 7.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.72 - 4.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.64542046554351810210877514561, −10.68800947925232910867151656788, −9.975345750685042932869262788575, −9.406972162772423037542273895924, −8.344987019065714352711608758357, −7.27587488863142301077681304759, −5.74275248813599601939656920538, −5.11966381803894759968886740987, −2.90131909042189896694729183546, −2.37216879248187225525086928102, 1.91518992126604725288393608900, 2.97928918284882633471542974877, 4.66193067297968710241226396641, 6.03095258388574183057348070191, 7.08694939090454121484626788493, 8.029647256940026562546331668832, 9.431583876398497404022020845464, 9.842262753692385099835138197714, 10.62242551627666239931356932913, 12.38688992765080214565517345855

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