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Properties

Label 2-252-36.11-c1-0-32
Degree $2$
Conductor $252$
Sign $-0.391 + 0.920i$
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  + (0.498 − 1.32i)2-s + (1.24 − 1.20i)3-s + (−1.50 − 1.31i)4-s + (2.06 − 1.19i)5-s + (−0.976 − 2.24i)6-s + (0.866 + 0.5i)7-s + (−2.49 + 1.33i)8-s + (0.0909 − 2.99i)9-s + (−0.549 − 3.33i)10-s + (−3.21 + 5.57i)11-s + (−3.45 + 0.171i)12-s + (2.15 + 3.72i)13-s + (1.09 − 0.896i)14-s + (1.13 − 3.97i)15-s + (0.516 + 3.96i)16-s − 1.89i·17-s + ⋯
L(s)  = 1  + (0.352 − 0.935i)2-s + (0.717 − 0.696i)3-s + (−0.751 − 0.659i)4-s + (0.925 − 0.534i)5-s + (−0.398 − 0.917i)6-s + (0.327 + 0.188i)7-s + (−0.882 + 0.470i)8-s + (0.0303 − 0.999i)9-s + (−0.173 − 1.05i)10-s + (−0.970 + 1.68i)11-s + (−0.998 + 0.0495i)12-s + (0.596 + 1.03i)13-s + (0.292 − 0.239i)14-s + (0.292 − 1.02i)15-s + (0.129 + 0.991i)16-s − 0.459i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03799 - 1.56906i\)
\(L(\frac12)\) \(\approx\) \(1.03799 - 1.56906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.498 + 1.32i)T \)
3 \( 1 + (-1.24 + 1.20i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-2.06 + 1.19i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.21 - 5.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.15 - 3.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.89iT - 17T^{2} \)
19 \( 1 + 0.529iT - 19T^{2} \)
23 \( 1 + (2.64 + 4.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.301 + 0.174i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.09 - 2.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.842T + 37T^{2} \)
41 \( 1 + (-4.58 + 2.64i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.38 - 4.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.04 - 3.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.5iT - 53T^{2} \)
59 \( 1 + (2.66 + 4.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.49 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.32 + 5.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.12T + 71T^{2} \)
73 \( 1 - 5.07T + 73T^{2} \)
79 \( 1 + (-2.87 - 1.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.23 + 5.60i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 + (3.86 - 6.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.10120981660713913805244663820, −10.83394498346426865629501207123, −9.612767366110850287269707155252, −9.213472354281144238001825135098, −8.022160021784305027838413632342, −6.65983915922805903557312768561, −5.34276449994880976160557677269, −4.24896739166462908020907610937, −2.43956377609209318051417279626, −1.69556572526630379711825087417, 2.83164691527836482956043705120, 3.84625527858788310969138459972, 5.51243989388606735013059063883, 5.91273199925593935658044115992, 7.66291887379418049755345017830, 8.274443017803975164520580113182, 9.278176783806706290871521191011, 10.37526557421555253669622440413, 11.05362133774369898346961965345, 12.90280223554906803496821625165

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