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Properties

Label 2-252-63.16-c1-0-4
Degree $2$
Conductor $252$
Sign $-0.118 + 0.992i$
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  + (−1.58 + 0.705i)3-s − 2.52·5-s + (1.98 − 1.75i)7-s + (2.00 − 2.23i)9-s − 1.37·11-s + (−2.80 − 4.84i)13-s + (3.98 − 1.77i)15-s + (−2.69 − 4.66i)17-s + (2.44 − 4.23i)19-s + (−1.89 + 4.17i)21-s + 4.17·23-s + 1.35·25-s + (−1.59 + 4.94i)27-s + (−1.56 + 2.71i)29-s + (−2.40 + 4.15i)31-s + ⋯
L(s)  = 1  + (−0.913 + 0.407i)3-s − 1.12·5-s + (0.748 − 0.662i)7-s + (0.668 − 0.743i)9-s − 0.414·11-s + (−0.776 − 1.34i)13-s + (1.02 − 0.458i)15-s + (−0.653 − 1.13i)17-s + (0.561 − 0.972i)19-s + (−0.414 + 0.910i)21-s + 0.870·23-s + 0.270·25-s + (−0.307 + 0.951i)27-s + (−0.291 + 0.504i)29-s + (−0.431 + 0.746i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.118 + 0.992i$
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: \((2,\ 252,\ (\ :1/2),\ -0.118 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363494 - 0.409552i\)
\(L(\frac12)\) \(\approx\) \(0.363494 - 0.409552i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.705i)T \)
7 \( 1 + (-1.98 + 1.75i)T \)
good5 \( 1 + 2.52T + 5T^{2} \)
11 \( 1 + 1.37T + 11T^{2} \)
13 \( 1 + (2.80 + 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.69 + 4.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.44 + 4.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.17T + 23T^{2} \)
29 \( 1 + (1.56 - 2.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.40 - 4.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.69 - 4.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.02 + 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.44 + 4.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.82 - 4.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.00 + 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.13 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.42 - 5.93i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.05 + 7.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 + (-3.51 - 6.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.48 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.75 + 4.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.894 + 1.54i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.59138498380315285553078511552, −10.97719470839706870836742684041, −10.20411355271232403043711278177, −8.891596794646348703692489730032, −7.53373229016543189498062399786, −7.09042529630246462421576788939, −5.20832877589686829445954485681, −4.71237662517807881663302459565, −3.27697571360900662398503253846, −0.48593273287634244791655566828, 1.93769185386045341182781990301, 4.09893704049583998308688233132, 5.06066417297019758629033661638, 6.26621939756148790158076109720, 7.46313744420125458124983744347, 8.109124512449674986333055943721, 9.409271036306349103100696317946, 10.82999493702509683689652427451, 11.45271163918456052133289601266, 12.12116388319897806796969336790

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