Properties

Label 2-252-9.4-c1-0-2
Degree 22
Conductor 252252
Sign 0.9270.373i0.927 - 0.373i
Analytic cond. 2.012232.01223
Root an. cond. 1.418531.41853
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 − 0.272i)3-s + (−0.119 + 0.207i)5-s + (0.5 + 0.866i)7-s + (2.85 + 0.931i)9-s + (2.56 + 4.43i)11-s + (2.44 − 4.23i)13-s + (0.260 − 0.321i)15-s + 3.70·17-s + 3.66·19-s + (−0.619 − 1.61i)21-s + (−3.71 + 6.42i)23-s + (2.47 + 4.28i)25-s + (−4.62 − 2.36i)27-s + (−1.73 − 3.00i)29-s + (0.358 − 0.621i)31-s + ⋯
L(s)  = 1  + (−0.987 − 0.157i)3-s + (−0.0534 + 0.0926i)5-s + (0.188 + 0.327i)7-s + (0.950 + 0.310i)9-s + (0.772 + 1.33i)11-s + (0.677 − 1.17i)13-s + (0.0673 − 0.0830i)15-s + 0.898·17-s + 0.839·19-s + (−0.135 − 0.352i)21-s + (−0.773 + 1.34i)23-s + (0.494 + 0.856i)25-s + (−0.890 − 0.455i)27-s + (−0.321 − 0.557i)29-s + (0.0644 − 0.111i)31-s + ⋯

Functional equation

Λ(s)=(252s/2ΓC(s)L(s)=((0.9270.373i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(252s/2ΓC(s+1/2)L(s)=((0.9270.373i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    223272^{2} \cdot 3^{2} \cdot 7
Sign: 0.9270.373i0.927 - 0.373i
Analytic conductor: 2.012232.01223
Root analytic conductor: 1.418531.41853
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ252(85,)\chi_{252} (85, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :1/2), 0.9270.373i)(2,\ 252,\ (\ :1/2),\ 0.927 - 0.373i)

Particular Values

L(1)L(1) \approx 0.987593+0.191240i0.987593 + 0.191240i
L(12)L(\frac12) \approx 0.987593+0.191240i0.987593 + 0.191240i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(1.71+0.272i)T 1 + (1.71 + 0.272i)T
7 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(0.1190.207i)T+(2.54.33i)T2 1 + (0.119 - 0.207i)T + (-2.5 - 4.33i)T^{2}
11 1+(2.564.43i)T+(5.5+9.52i)T2 1 + (-2.56 - 4.43i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.44+4.23i)T+(6.511.2i)T2 1 + (-2.44 + 4.23i)T + (-6.5 - 11.2i)T^{2}
17 13.70T+17T2 1 - 3.70T + 17T^{2}
19 13.66T+19T2 1 - 3.66T + 19T^{2}
23 1+(3.716.42i)T+(11.519.9i)T2 1 + (3.71 - 6.42i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.73+3.00i)T+(14.5+25.1i)T2 1 + (1.73 + 3.00i)T + (-14.5 + 25.1i)T^{2}
31 1+(0.358+0.621i)T+(15.526.8i)T2 1 + (-0.358 + 0.621i)T + (-15.5 - 26.8i)T^{2}
37 14.60T+37T2 1 - 4.60T + 37T^{2}
41 1+(2.804.85i)T+(20.535.5i)T2 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2}
43 1+(6.24+10.8i)T+(21.5+37.2i)T2 1 + (6.24 + 10.8i)T + (-21.5 + 37.2i)T^{2}
47 1+(2.16+3.75i)T+(23.5+40.7i)T2 1 + (2.16 + 3.75i)T + (-23.5 + 40.7i)T^{2}
53 1+0.942T+53T2 1 + 0.942T + 53T^{2}
59 1+(3.786.56i)T+(29.551.0i)T2 1 + (3.78 - 6.56i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.75+4.77i)T+(30.5+52.8i)T2 1 + (2.75 + 4.77i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.330+0.571i)T+(33.558.0i)T2 1 + (-0.330 + 0.571i)T + (-33.5 - 58.0i)T^{2}
71 113.7T+71T2 1 - 13.7T + 71T^{2}
73 1+3.66T+73T2 1 + 3.66T + 73T^{2}
79 1+(3.115.39i)T+(39.5+68.4i)T2 1 + (-3.11 - 5.39i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.85+8.40i)T+(41.5+71.8i)T2 1 + (4.85 + 8.40i)T + (-41.5 + 71.8i)T^{2}
89 1+7.48T+89T2 1 + 7.48T + 89T^{2}
97 1+(8.5714.8i)T+(48.5+84.0i)T2 1 + (-8.57 - 14.8i)T + (-48.5 + 84.0i)T^{2}
show more
show less
   L(s)=p j=12(1αj,pps)1L(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

−11.97799351870408510048550690269, −11.37693140805465257400742687260, −10.17715003106063684072318641559, −9.527149679089405759578863469670, −7.930476536145668280255029103203, −7.12971908713605981445671675979, −5.88775049970322527640907444623, −5.09276305379219344796297682095, −3.63719118170593401263908072317, −1.49537085468508805979606334047, 1.13148452339232045201304986063, 3.60962090874277268982833436381, 4.69991105010304886346018262146, 6.02837776730201479444681698013, 6.67220598287954726968658860231, 8.090643612147532324380121976392, 9.178984024196770397496498408289, 10.25084022203673387344958845302, 11.22211114282473288925936106151, 11.74350293369536384169631102504

Graph of the ZZ-function along the critical line