↑

Properties

Label 2-3104-776.315-c0-0-0
Degree 22
Conductor 31043104
Sign −0.634−0.773i-0.634 - 0.773i
Analytic cond. 1.549091.54909
Root an. cond. 1.244621.24462
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 1.86i)3-s + (−2.36 + 1.36i)9-s + (1.67 + 0.448i)11-s + (0.0999 + 0.758i)17-s + (1.70 + 0.707i)19-s + (0.258 − 0.965i)25-s + (−2.36 − 2.36i)27-s + 3.34i·33-s + (−1.12 − 1.46i)41-s + (−0.448 − 0.258i)43-s + (−0.965 − 0.258i)49-s + (−1.36 + 0.565i)51-s + (−0.465 + 3.53i)57-s + (−0.758 − 0.0999i)59-s + (−1.12 − 0.465i)67-s + ⋯
L(s)  = 1  + (0.5 + 1.86i)3-s + (−2.36 + 1.36i)9-s + (1.67 + 0.448i)11-s + (0.0999 + 0.758i)17-s + (1.70 + 0.707i)19-s + (0.258 − 0.965i)25-s + (−2.36 − 2.36i)27-s + 3.34i·33-s + (−1.12 − 1.46i)41-s + (−0.448 − 0.258i)43-s + (−0.965 − 0.258i)49-s + (−1.36 + 0.565i)51-s + (−0.465 + 3.53i)57-s + (−0.758 − 0.0999i)59-s + (−1.12 − 0.465i)67-s + ⋯

Functional equation

Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.634−0.773i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3104s/2 ΓC(s) L(s)=((−0.634−0.773i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31043104    =    25⋅972^{5} \cdot 97
Sign: −0.634−0.773i-0.634 - 0.773i
Analytic conductor: 1.549091.54909
Root analytic conductor: 1.244621.24462
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3104(2255,⋅)\chi_{3104} (2255, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3104, ( :0), −0.634−0.773i)(2,\ 3104,\ (\ :0),\ -0.634 - 0.773i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5703482251.570348225
L(12)L(\frac12) ≈\approx 1.5703482251.570348225
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
97 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
good3 1+(−0.5−1.86i)T+(−0.866+0.5i)T2 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2}
5 1+(−0.258+0.965i)T2 1 + (-0.258 + 0.965i)T^{2}
7 1+(0.965+0.258i)T2 1 + (0.965 + 0.258i)T^{2}
11 1+(−1.67−0.448i)T+(0.866+0.5i)T2 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2}
13 1+(−0.258+0.965i)T2 1 + (-0.258 + 0.965i)T^{2}
17 1+(−0.0999−0.758i)T+(−0.965+0.258i)T2 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2}
19 1+(−1.70−0.707i)T+(0.707+0.707i)T2 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2}
23 1+(−0.965+0.258i)T2 1 + (-0.965 + 0.258i)T^{2}
29 1+(0.258−0.965i)T2 1 + (0.258 - 0.965i)T^{2}
31 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
37 1+(−0.965−0.258i)T2 1 + (-0.965 - 0.258i)T^{2}
41 1+(1.12+1.46i)T+(−0.258+0.965i)T2 1 + (1.12 + 1.46i)T + (-0.258 + 0.965i)T^{2}
43 1+(0.448+0.258i)T+(0.5+0.866i)T2 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2}
47 1+T2 1 + T^{2}
53 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
59 1+(0.758+0.0999i)T+(0.965+0.258i)T2 1 + (0.758 + 0.0999i)T + (0.965 + 0.258i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(1.12+0.465i)T+(0.707+0.707i)T2 1 + (1.12 + 0.465i)T + (0.707 + 0.707i)T^{2}
71 1+(0.258+0.965i)T2 1 + (0.258 + 0.965i)T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1+iT2 1 + iT^{2}
83 1+(−0.258−1.96i)T+(−0.965+0.258i)T2 1 + (-0.258 - 1.96i)T + (-0.965 + 0.258i)T^{2}
89 1+(0.366−0.366i)T−iT2 1 + (0.366 - 0.366i)T - iT^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−9.296470997866164217582457396041, −8.615142044406262418226876743951, −7.969559635710770951515828098882, −6.85361436836872637708646030192, −5.89379660048272708510217606567, −5.14608106372624727472601955117, −4.31359786265935274662880606055, −3.71975299170670309814332859730, −3.10560984405979842812377444113, −1.74121614566766929558505231872, 1.02722502964051098795050805574, 1.62508712363600844694026457862, 3.00520405473588119922837482821, 3.35457065475229030563580767500, 4.88601603173039486235155587288, 5.89768748844911794249348819119, 6.55615746911265343052906834472, 7.15485321638792286962436810345, 7.67168897092172108011427864040, 8.554885493028798666649742014165

Graph of the ZZ-function along the critical line