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Properties

Label 2-252-28.27-c1-0-15
Degree $2$
Conductor $252$
Sign $-0.661 + 0.750i$
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  + (0.5 − 1.32i)2-s + (−1.50 − 1.32i)4-s − 2.64i·7-s + (−2.50 + 1.32i)8-s − 5.29i·11-s + (−3.50 − 1.32i)14-s + (0.500 + 3.96i)16-s + (−7.00 − 2.64i)22-s + 5.29i·23-s + 5·25-s + (−3.50 + 3.96i)28-s + 2·29-s + (5.50 + 1.32i)32-s + 6·37-s + 5.29i·43-s + (−7.00 + 7.93i)44-s + ⋯
L(s)  = 1  + (0.353 − 0.935i)2-s + (−0.750 − 0.661i)4-s − 0.999i·7-s + (−0.883 + 0.467i)8-s − 1.59i·11-s + (−0.935 − 0.353i)14-s + (0.125 + 0.992i)16-s + (−1.49 − 0.564i)22-s + 1.10i·23-s + 25-s + (−0.661 + 0.749i)28-s + 0.371·29-s + (0.972 + 0.233i)32-s + 0.986·37-s + 0.806i·43-s + (−1.05 + 1.19i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.661 + 0.750i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.661 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528208 - 1.17011i\)
\(L(\frac12)\) \(\approx\) \(0.528208 - 1.17011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 1.32i)T \)
3 \( 1 \)
7 \( 1 + 2.64iT \)
good5 \( 1 - 5T^{2} \)
11 \( 1 + 5.29iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.29iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 5.29iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 15.8iT - 67T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.44873823039463598382753223712, −10.96014345974379194782959146475, −10.03384630134626803101284451735, −8.990776813701639835110735773678, −7.944170599950724135117929823773, −6.45656284749156489693295988644, −5.30825754932605045507871126943, −4.01509482432327481739222024615, −3.01526999505069296776811776982, −0.998177539737668739040491763176, 2.58754309290365295518166661006, 4.32704436396164722006082050468, 5.23447044052488790323012056384, 6.41497312201832572772558133114, 7.30327994275901525197194475622, 8.447308015944748974716475388576, 9.247886603319706769364924249349, 10.27664416446270182937686146892, 11.87554326925152230221395520467, 12.48581193760422415313143247331

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