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Properties

Label 2-3040-3040.949-c0-0-3
Degree 22
Conductor 30403040
Sign 0.471−0.881i0.471 - 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

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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: 0.471−0.881i0.471 - 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 1.8638636251.863863625
L(12)L(\frac12) ≈\approx 1.8638636251.863863625
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}
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   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

−8.875084113698747895873089015063, −8.405059755971622569785303140780, −7.17698316200044613455468719574, −6.48636975555224171645684679263, −5.90545989880627795759039049971, −5.13749417801363517164258482494, −4.43867462456293721666133590885, −4.02146312378638022803487195060, −2.51651465628986762326535853705, −1.49629673416958868005094115038, 1.03005769980918105566605803683, 2.27971281519166682233513556309, 3.31133773199288018303229731885, 3.81756865684891612562796899493, 5.28695287708505383061402091816, 5.71235312704180918727632779015, 6.26065768935395910974721392236, 6.92123129038630338568334020808, 7.73171500138131488415179915006, 8.714659902840706230896294688921

Graph of the ZZ-function along the critical line