Properties

Label 2-2548-13.3-c1-0-7
Degree 22
Conductor 25482548
Sign 0.01280.999i0.0128 - 0.999i
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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  + 5-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s + (−3.5 + 0.866i)13-s + (−1.5 + 2.59i)17-s + (−3 + 5.19i)19-s + (2 + 3.46i)23-s − 4·25-s + (3.5 + 6.06i)29-s − 4·31-s + (−4.5 − 7.79i)37-s + (4.5 + 7.79i)41-s + (−5 + 8.66i)43-s + (1.5 − 2.59i)45-s + 2·47-s + ⋯
L(s)  = 1  + 0.447·5-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + (−0.970 + 0.240i)13-s + (−0.363 + 0.630i)17-s + (−0.688 + 1.19i)19-s + (0.417 + 0.722i)23-s − 0.800·25-s + (0.649 + 1.12i)29-s − 0.718·31-s + (−0.739 − 1.28i)37-s + (0.702 + 1.21i)41-s + (−0.762 + 1.32i)43-s + (0.223 − 0.387i)45-s + 0.291·47-s + ⋯

Functional equation

Λ(s)=(2548s/2ΓC(s)L(s)=((0.01280.999i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2ΓC(s+1/2)L(s)=((0.01280.999i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    2272132^{2} \cdot 7^{2} \cdot 13
Sign: 0.01280.999i0.0128 - 0.999i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(393,)\chi_{2548} (393, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), 0.01280.999i)(2,\ 2548,\ (\ :1/2),\ 0.0128 - 0.999i)

Particular Values

L(1)L(1) \approx 1.4286280381.428628038
L(12)L(\frac12) \approx 1.4286280381.428628038
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 1
13 1+(3.50.866i)T 1 + (3.5 - 0.866i)T
good3 1+(1.5+2.59i)T2 1 + (-1.5 + 2.59i)T^{2}
5 1T+5T2 1 - T + 5T^{2}
11 1+(11.73i)T+(5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(1.52.59i)T+(8.514.7i)T2 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(35.19i)T+(9.516.4i)T2 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2}
23 1+(23.46i)T+(11.5+19.9i)T2 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.56.06i)T+(14.5+25.1i)T2 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+(4.5+7.79i)T+(18.5+32.0i)T2 1 + (4.5 + 7.79i)T + (-18.5 + 32.0i)T^{2}
41 1+(4.57.79i)T+(20.5+35.5i)T2 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2}
43 1+(58.66i)T+(21.537.2i)T2 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2}
47 12T+47T2 1 - 2T + 47T^{2}
53 19T+53T2 1 - 9T + 53T^{2}
59 1+(7+12.1i)T+(29.551.0i)T2 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.54.33i)T+(30.552.8i)T2 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(46.92i)T+(33.5+58.0i)T2 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2}
71 1+(58.66i)T+(35.561.4i)T2 1 + (5 - 8.66i)T + (-35.5 - 61.4i)T^{2}
73 17T+73T2 1 - 7T + 73T^{2}
79 12T+79T2 1 - 2T + 79T^{2}
83 16T+83T2 1 - 6T + 83T^{2}
89 1+(35.19i)T+(44.5+77.0i)T2 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}
97 1+(1+1.73i)T+(48.584.0i)T2 1 + (-1 + 1.73i)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

−9.237345327956482721394767972710, −8.392251054166184215278769066262, −7.43196520287299727545806214717, −6.79723032020176407291172815408, −6.07640451188673910534696769876, −5.20859730866804991322825135776, −4.21834833239435695751417974752, −3.55765056546726034712774659494, −2.22336974564856587144707062007, −1.38768714962326992062310427925, 0.46032145690407628496447076092, 2.07222224257763937184449827180, 2.64698627841450364886970071717, 3.99833159966872107770044250378, 4.85239784626783516568776957328, 5.44206600500674848502534607058, 6.54261187815172419624591523247, 7.11196513489318938581663957820, 7.931956435518194900755251474421, 8.797434169369425914973336238245

Graph of the ZZ-function along the critical line