Properties

Label 2-532-133.121-c1-0-13
Degree 22
Conductor 532532
Sign 0.912+0.409i-0.912 + 0.409i
Analytic cond. 4.248044.24804
Root an. cond. 2.061072.06107
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  + (1.02 − 1.77i)3-s − 2.32·5-s + (−0.804 − 2.52i)7-s + (−0.609 − 1.05i)9-s + (−1.44 − 2.49i)11-s + (−0.126 + 0.219i)13-s + (−2.38 + 4.13i)15-s + (−2.14 + 3.71i)17-s + (−3.08 − 3.07i)19-s + (−5.30 − 1.15i)21-s + (−0.926 − 1.60i)23-s + 0.399·25-s + 3.65·27-s + (0.114 − 0.199i)29-s + (−4.46 − 7.73i)31-s + ⋯
L(s)  = 1  + (0.592 − 1.02i)3-s − 1.03·5-s + (−0.303 − 0.952i)7-s + (−0.203 − 0.351i)9-s + (−0.434 − 0.752i)11-s + (−0.0350 + 0.0607i)13-s + (−0.616 + 1.06i)15-s + (−0.519 + 0.900i)17-s + (−0.707 − 0.706i)19-s + (−1.15 − 0.252i)21-s + (−0.193 − 0.334i)23-s + 0.0798·25-s + 0.704·27-s + (0.0213 − 0.0369i)29-s + (−0.801 − 1.38i)31-s + ⋯

Functional equation

Λ(s)=(532s/2ΓC(s)L(s)=((0.912+0.409i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(532s/2ΓC(s+1/2)L(s)=((0.912+0.409i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 532532    =    227192^{2} \cdot 7 \cdot 19
Sign: 0.912+0.409i-0.912 + 0.409i
Analytic conductor: 4.248044.24804
Root analytic conductor: 2.061072.06107
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ532(121,)\chi_{532} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 532, ( :1/2), 0.912+0.409i)(2,\ 532,\ (\ :1/2),\ -0.912 + 0.409i)

Particular Values

L(1)L(1) \approx 0.2021600.943319i0.202160 - 0.943319i
L(12)L(\frac12) \approx 0.2021600.943319i0.202160 - 0.943319i
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
7 1+(0.804+2.52i)T 1 + (0.804 + 2.52i)T
19 1+(3.08+3.07i)T 1 + (3.08 + 3.07i)T
good3 1+(1.02+1.77i)T+(1.52.59i)T2 1 + (-1.02 + 1.77i)T + (-1.5 - 2.59i)T^{2}
5 1+2.32T+5T2 1 + 2.32T + 5T^{2}
11 1+(1.44+2.49i)T+(5.5+9.52i)T2 1 + (1.44 + 2.49i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.1260.219i)T+(6.511.2i)T2 1 + (0.126 - 0.219i)T + (-6.5 - 11.2i)T^{2}
17 1+(2.143.71i)T+(8.514.7i)T2 1 + (2.14 - 3.71i)T + (-8.5 - 14.7i)T^{2}
23 1+(0.926+1.60i)T+(11.5+19.9i)T2 1 + (0.926 + 1.60i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.114+0.199i)T+(14.525.1i)T2 1 + (-0.114 + 0.199i)T + (-14.5 - 25.1i)T^{2}
31 1+(4.46+7.73i)T+(15.5+26.8i)T2 1 + (4.46 + 7.73i)T + (-15.5 + 26.8i)T^{2}
37 1+(3.34+5.80i)T+(18.532.0i)T2 1 + (-3.34 + 5.80i)T + (-18.5 - 32.0i)T^{2}
41 1+(5.509.53i)T+(20.5+35.5i)T2 1 + (-5.50 - 9.53i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.093.62i)T+(21.5+37.2i)T2 1 + (-2.09 - 3.62i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.61+7.99i)T+(23.5+40.7i)T2 1 + (4.61 + 7.99i)T + (-23.5 + 40.7i)T^{2}
53 18.63T+53T2 1 - 8.63T + 53T^{2}
59 1+(0.3510.608i)T+(29.551.0i)T2 1 + (0.351 - 0.608i)T + (-29.5 - 51.0i)T^{2}
61 1+(5.19+8.98i)T+(30.5+52.8i)T2 1 + (5.19 + 8.98i)T + (-30.5 + 52.8i)T^{2}
67 110.1T+67T2 1 - 10.1T + 67T^{2}
71 1+(1.57+2.72i)T+(35.5+61.4i)T2 1 + (1.57 + 2.72i)T + (-35.5 + 61.4i)T^{2}
73 1+(3.48+6.03i)T+(36.563.2i)T2 1 + (-3.48 + 6.03i)T + (-36.5 - 63.2i)T^{2}
79 115.9T+79T2 1 - 15.9T + 79T^{2}
83 15.34T+83T2 1 - 5.34T + 83T^{2}
89 1+(1.532.66i)T+(44.5+77.0i)T2 1 + (-1.53 - 2.66i)T + (-44.5 + 77.0i)T^{2}
97 1+(6.80+11.7i)T+(48.5+84.0i)T2 1 + (6.80 + 11.7i)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

−10.77190909612069317551444660071, −9.427073963706286224619922104553, −8.219035205876954078239487341131, −7.908900981269931580130792961496, −7.01234797174486410887663575251, −6.16587615637386606697608460449, −4.45588020045093140429281636415, −3.57959478931039603925445838423, −2.25459845731733058026308325112, −0.50714348904975422759098615716, 2.45609162453092156430452731459, 3.55654279440074419731740476251, 4.41852493561199756847949320626, 5.39353633202998682681040387242, 6.80415577368535854649862277613, 7.83504756968586306380081868369, 8.724139142375890055864385483960, 9.380960180921460277208555307885, 10.22248624328120474478790531564, 11.12001732857196545225460228898

Graph of the ZZ-function along the critical line