↑

Properties

Label 2-3040-3040.949-c0-0-5
Degree 22
Conductor 30403040
Sign −0.471+0.881i-0.471 + 0.881i
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  + (−0.956 − 0.290i)2-s + (0.871 − 0.360i)3-s + (0.831 + 0.555i)4-s + (0.382 − 0.923i)5-s + (−0.938 + 0.0924i)6-s + (−0.634 − 0.773i)8-s + (−0.0785 + 0.0785i)9-s + (−0.634 + 0.773i)10-s + (0.360 + 0.149i)11-s + (0.924 + 0.183i)12-s + (−0.761 − 1.83i)13-s − 0.942i·15-s + (0.382 + 0.923i)16-s + (0.0980 − 0.0523i)18-s + (−0.382 − 0.923i)19-s + (0.831 − 0.555i)20-s + ⋯
L(s)  = 1  + (−0.956 − 0.290i)2-s + (0.871 − 0.360i)3-s + (0.831 + 0.555i)4-s + (0.382 − 0.923i)5-s + (−0.938 + 0.0924i)6-s + (−0.634 − 0.773i)8-s + (−0.0785 + 0.0785i)9-s + (−0.634 + 0.773i)10-s + (0.360 + 0.149i)11-s + (0.924 + 0.183i)12-s + (−0.761 − 1.83i)13-s − 0.942i·15-s + (0.382 + 0.923i)16-s + (0.0980 − 0.0523i)18-s + (−0.382 − 0.923i)19-s + (0.831 − 0.555i)20-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.471+0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((−0.471+0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: −0.471+0.881i-0.471 + 0.881i
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), −0.471+0.881i)(2,\ 3040,\ (\ :0),\ -0.471 + 0.881i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0103867631.010386763
L(12)L(\frac12) ≈\approx 1.0103867631.010386763
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+(0.956+0.290i)T 1 + (0.956 + 0.290i)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+(−0.871+0.360i)T+(0.707−0.707i)T2 1 + (-0.871 + 0.360i)T + (0.707 - 0.707i)T^{2}
7 1−iT2 1 - iT^{2}
11 1+(−0.360−0.149i)T+(0.707+0.707i)T2 1 + (-0.360 - 0.149i)T + (0.707 + 0.707i)T^{2}
13 1+(0.761+1.83i)T+(−0.707+0.707i)T2 1 + (0.761 + 1.83i)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+(−0.222+0.536i)T+(−0.707−0.707i)T2 1 + (-0.222 + 0.536i)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+(1.62+0.674i)T+(0.707+0.707i)T2 1 + (1.62 + 0.674i)T + (0.707 + 0.707i)T^{2}
59 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
61 1+(−1.81+0.750i)T+(0.707−0.707i)T2 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2}
67 1+(−1.42+0.591i)T+(0.707−0.707i)T2 1 + (-1.42 + 0.591i)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 1−0.196T+T2 1 - 0.196T + 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

−8.593463689758460033649765500812, −8.066726660139136723864060245774, −7.58167513855208584196958301240, −6.64906229916383202686920594385, −5.63982490462760850829577538526, −4.82569316982517014941965042666, −3.52509084678883565088301187815, −2.65964146097347154112870946976, −1.97702351905523736038437915001, −0.72568875372065699814535023896, 1.77073380192371602501402105405, 2.43291956416617847574705458521, 3.38808933503803121215862376225, 4.30908113610089628417178114359, 5.61331862493034753561104382441, 6.49035220494985368783869437941, 6.87692482626891387280822057976, 7.77125298373777711846309464390, 8.529182075997102467757902797723, 9.163980795859505787574810193008

Graph of the ZZ-function along the critical line