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Properties

Label 2-252-28.3-c1-0-12
Degree $2$
Conductor $252$
Sign $-0.196 + 0.980i$
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  + (−0.895 + 1.09i)2-s + (−0.395 − 1.96i)4-s + (−1.5 + 0.866i)5-s + (−2.29 − 1.32i)7-s + (2.49 + 1.32i)8-s + (0.395 − 2.41i)10-s + (−2.29 − 1.32i)11-s − 3.46i·13-s + (3.49 − 1.32i)14-s + (−3.68 + 1.55i)16-s + (−6 − 3.46i)17-s + (2.29 + 2.59i)20-s + (3.49 − 1.32i)22-s + (4.58 − 2.64i)23-s + (−1 + 1.73i)25-s + (3.79 + 3.10i)26-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)2-s + (−0.197 − 0.980i)4-s + (−0.670 + 0.387i)5-s + (−0.866 − 0.499i)7-s + (0.883 + 0.467i)8-s + (0.125 − 0.764i)10-s + (−0.690 − 0.398i)11-s − 0.960i·13-s + (0.935 − 0.353i)14-s + (−0.921 + 0.387i)16-s + (−1.45 − 0.840i)17-s + (0.512 + 0.580i)20-s + (0.746 − 0.282i)22-s + (0.955 − 0.551i)23-s + (−0.200 + 0.346i)25-s + (0.743 + 0.608i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.196 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165790 - 0.202297i\)
\(L(\frac12)\) \(\approx\) \(0.165790 - 0.202297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.895 - 1.09i)T \)
3 \( 1 \)
7 \( 1 + (2.29 + 1.32i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.29 + 1.32i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.58 + 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-2.29 + 3.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 + (4.58 + 7.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.5 + 6.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.87 - 11.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9 - 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.87 + 3.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.58T + 83T^{2} \)
89 \( 1 + (-9 + 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−11.38610109112951576081613333247, −10.72395262675180686649635653506, −9.783987260322882356926020100346, −8.766920788941284831168553052464, −7.68383187160324922890331287662, −6.99821487854827112175172527102, −5.92912862406213373160656705118, −4.58612774939956256033318742322, −2.97472740750053989110719882266, −0.24107456398891390177635877369, 2.14874943459000897578375773083, 3.60773835850154606200124063080, 4.73585417859642499420512598767, 6.53963723115059543675804966223, 7.61005937844386687749653414619, 8.767822640030567965475869603707, 9.287888959978556380010103415954, 10.47347937327041214134561028020, 11.32703357132728112514933682356, 12.28133681627459649055463061768

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