↑

Properties

Label 2-3040-3040.949-c0-0-1
Degree 22
Conductor 30403040
Sign −0.956+0.290i-0.956 + 0.290i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
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  + (0.471 + 0.881i)2-s + (−0.536 + 0.222i)3-s + (−0.555 + 0.831i)4-s + (−0.382 + 0.923i)5-s + (−0.448 − 0.368i)6-s + (−0.995 − 0.0980i)8-s + (−0.468 + 0.468i)9-s + (−0.995 + 0.0980i)10-s + (1.81 + 0.750i)11-s + (0.113 − 0.569i)12-s + (0.591 + 1.42i)13-s − 0.580i·15-s + (−0.382 − 0.923i)16-s + (−0.634 − 0.192i)18-s + (0.382 + 0.923i)19-s + (−0.555 − 0.831i)20-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)2-s + (−0.536 + 0.222i)3-s + (−0.555 + 0.831i)4-s + (−0.382 + 0.923i)5-s + (−0.448 − 0.368i)6-s + (−0.995 − 0.0980i)8-s + (−0.468 + 0.468i)9-s + (−0.995 + 0.0980i)10-s + (1.81 + 0.750i)11-s + (0.113 − 0.569i)12-s + (0.591 + 1.42i)13-s − 0.580i·15-s + (−0.382 − 0.923i)16-s + (−0.634 − 0.192i)18-s + (0.382 + 0.923i)19-s + (−0.555 − 0.831i)20-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.956+0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: −0.956+0.290i-0.956 + 0.290i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(949,⋅)\chi_{3040} (949, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3040, ( :0), −0.956+0.290i)(2,\ 3040,\ (\ :0),\ -0.956 + 0.290i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1315119721.131511972
L(12)L(\frac12) ≈\approx 1.1315119721.131511972
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+(−0.471−0.881i)T 1 + (-0.471 - 0.881i)T
5 1+(0.382−0.923i)T 1 + (0.382 - 0.923i)T
19 1+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
good3 1+(0.536−0.222i)T+(0.707−0.707i)T2 1 + (0.536 - 0.222i)T + (0.707 - 0.707i)T^{2}
7 1−iT2 1 - iT^{2}
11 1+(−1.81−0.750i)T+(0.707+0.707i)T2 1 + (-1.81 - 0.750i)T + (0.707 + 0.707i)T^{2}
13 1+(−0.591−1.42i)T+(−0.707+0.707i)T2 1 + (-0.591 - 1.42i)T + (-0.707 + 0.707i)T^{2}
17 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
31 1−T2 1 - T^{2}
37 1+(−0.674+1.62i)T+(−0.707−0.707i)T2 1 + (-0.674 + 1.62i)T + (-0.707 - 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(1.76+0.732i)T+(0.707+0.707i)T2 1 + (1.76 + 0.732i)T + (0.707 + 0.707i)T^{2}
59 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
61 1+(0.360−0.149i)T+(0.707−0.707i)T2 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2}
67 1+(0.181−0.0750i)T+(0.707−0.707i)T2 1 + (0.181 - 0.0750i)T + (0.707 - 0.707i)T^{2}
71 1−iT2 1 - iT^{2}
73 1+iT2 1 + iT^{2}
79 1+T2 1 + T^{2}
83 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
89 1−iT2 1 - iT^{2}
97 1+1.26T+T2 1 + 1.26T + 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

−9.266247214721606424274278011987, −8.424042360580877809345696405946, −7.55220497763297888215987764063, −6.90705156477841191964613050333, −6.27218789650653308565894848158, −5.79469626047336333373769777045, −4.48198817898417787544505424680, −4.13029898749270115987058702410, −3.26115164921092015052701762967, −1.88576112041199998146531585560, 0.74025334897137071275010847702, 1.32532505189147106411326753185, 3.07588300972470452744340532686, 3.59033774176315105235661203267, 4.56225085477445754843379378836, 5.33877021886169900103624505017, 6.07138773136718411965360601303, 6.59429381753107160082584367005, 7.976238573384390096896895022004, 8.766450460323700511248448641118

Graph of the ZZ-function along the critical line