↑

Properties

Label 2-3248-3248.1371-c0-0-1
Degree 22
Conductor 32483248
Sign −0.724+0.689i-0.724 + 0.689i
Analytic cond. 1.620961.62096
Root an. cond. 1.273171.27317
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  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.623 − 0.781i)7-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)9-s + (0.0990 + 0.433i)11-s + (0.433 + 0.900i)14-s + (0.623 + 0.781i)16-s + (−0.433 + 0.900i)18-s − 0.445i·22-s + (−1.40 − 0.678i)23-s + (−0.781 − 0.623i)25-s + (−0.222 − 0.974i)28-s + (0.781 + 0.623i)29-s + (−0.433 − 0.900i)32-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.623 − 0.781i)7-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)9-s + (0.0990 + 0.433i)11-s + (0.433 + 0.900i)14-s + (0.623 + 0.781i)16-s + (−0.433 + 0.900i)18-s − 0.445i·22-s + (−1.40 − 0.678i)23-s + (−0.781 − 0.623i)25-s + (−0.222 − 0.974i)28-s + (0.781 + 0.623i)29-s + (−0.433 − 0.900i)32-s + ⋯

Functional equation

Λ(s)=(3248s/2 ΓC(s) L(s)=((−0.724+0.689i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3248s/2 ΓC(s) L(s)=((−0.724+0.689i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32483248    =    24⋅7⋅292^{4} \cdot 7 \cdot 29
Sign: −0.724+0.689i-0.724 + 0.689i
Analytic conductor: 1.620961.62096
Root analytic conductor: 1.273171.27317
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3248(1371,⋅)\chi_{3248} (1371, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3248, ( :0), −0.724+0.689i)(2,\ 3248,\ (\ :0),\ -0.724 + 0.689i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.48573022710.4857302271
L(12)L(\frac12) ≈\approx 0.48573022710.4857302271
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+(0.974+0.222i)T 1 + (0.974 + 0.222i)T
7 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
29 1+(−0.781−0.623i)T 1 + (-0.781 - 0.623i)T
good3 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
5 1+(0.781+0.623i)T2 1 + (0.781 + 0.623i)T^{2}
11 1+(−0.0990−0.433i)T+(−0.900+0.433i)T2 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2}
13 1+(0.433+0.900i)T2 1 + (0.433 + 0.900i)T^{2}
17 1−iT2 1 - iT^{2}
19 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
23 1+(1.40+0.678i)T+(0.623+0.781i)T2 1 + (1.40 + 0.678i)T + (0.623 + 0.781i)T^{2}
31 1+(0.781+0.623i)T2 1 + (0.781 + 0.623i)T^{2}
37 1+(−0.400+1.75i)T+(−0.900−0.433i)T2 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2}
41 1−iT2 1 - iT^{2}
43 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
47 1+(0.433+0.900i)T2 1 + (0.433 + 0.900i)T^{2}
53 1+(1.00+0.351i)T+(0.781+0.623i)T2 1 + (1.00 + 0.351i)T + (0.781 + 0.623i)T^{2}
59 1+iT2 1 + iT^{2}
61 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
67 1+(0.119+0.189i)T+(−0.433+0.900i)T2 1 + (0.119 + 0.189i)T + (-0.433 + 0.900i)T^{2}
71 1+(1.75−0.400i)T+(0.900−0.433i)T2 1 + (1.75 - 0.400i)T + (0.900 - 0.433i)T^{2}
73 1+(0.781−0.623i)T2 1 + (0.781 - 0.623i)T^{2}
79 1+(0.566+0.900i)T+(−0.433+0.900i)T2 1 + (0.566 + 0.900i)T + (-0.433 + 0.900i)T^{2}
83 1+(−0.974+0.222i)T2 1 + (-0.974 + 0.222i)T^{2}
89 1+(0.781+0.623i)T2 1 + (0.781 + 0.623i)T^{2}
97 1+(0.974−0.222i)T2 1 + (0.974 - 0.222i)T^{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

−8.675137691603261588530466043675, −7.78990011369042505901492036814, −7.18780971124149122140150718675, −6.43607425107033928361535750204, −5.95871492927116247624757796466, −4.34467127100978390367589275376, −3.76019273938711597013411813092, −2.79751149489010250808362642237, −1.67013315026869209862449704218, −0.39269000606397556428996398437, 1.54520592885330843561989647062, 2.45312071749943158130974069066, 3.32825110508361389622980558004, 4.63499297496160172683160183051, 5.70773938991581841135505496232, 6.08125713895557979743950727664, 6.98558629915933085173009152538, 7.889874896906889848316399528747, 8.253559737825681082815250214471, 9.099371413813116497137421282410

Graph of the ZZ-function along the critical line