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Properties

Label 2-3840-3840.2309-c0-0-0
Degree 22
Conductor 38403840
Sign 0.985+0.170i0.985 + 0.170i
Analytic cond. 1.916401.91640
Root an. cond. 1.384341.38434
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  + (−0.903 + 0.427i)2-s + (0.146 − 0.989i)3-s + (0.634 − 0.773i)4-s + (−0.998 − 0.0490i)5-s + (0.290 + 0.956i)6-s + (−0.242 + 0.970i)8-s + (−0.956 − 0.290i)9-s + (0.923 − 0.382i)10-s + (−0.671 − 0.740i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (0.289 + 1.45i)17-s + (0.989 − 0.146i)18-s + (0.207 + 0.439i)19-s + (−0.671 + 0.740i)20-s + ⋯
L(s)  = 1  + (−0.903 + 0.427i)2-s + (0.146 − 0.989i)3-s + (0.634 − 0.773i)4-s + (−0.998 − 0.0490i)5-s + (0.290 + 0.956i)6-s + (−0.242 + 0.970i)8-s + (−0.956 − 0.290i)9-s + (0.923 − 0.382i)10-s + (−0.671 − 0.740i)12-s + (−0.195 + 0.980i)15-s + (−0.195 − 0.980i)16-s + (0.289 + 1.45i)17-s + (0.989 − 0.146i)18-s + (0.207 + 0.439i)19-s + (−0.671 + 0.740i)20-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=((0.985+0.170i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s) L(s)=((0.985+0.170i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: 0.985+0.170i0.985 + 0.170i
Analytic conductor: 1.916401.91640
Root analytic conductor: 1.384341.38434
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3840(2309,⋅)\chi_{3840} (2309, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3840, ( :0), 0.985+0.170i)(2,\ 3840,\ (\ :0),\ 0.985 + 0.170i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.66042171500.6604217150
L(12)L(\frac12) ≈\approx 0.66042171500.6604217150
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+(0.903−0.427i)T 1 + (0.903 - 0.427i)T
3 1+(−0.146+0.989i)T 1 + (-0.146 + 0.989i)T
5 1+(0.998+0.0490i)T 1 + (0.998 + 0.0490i)T
good7 1+(−0.555−0.831i)T2 1 + (-0.555 - 0.831i)T^{2}
11 1+(0.471−0.881i)T2 1 + (0.471 - 0.881i)T^{2}
13 1+(0.0980+0.995i)T2 1 + (0.0980 + 0.995i)T^{2}
17 1+(−0.289−1.45i)T+(−0.923+0.382i)T2 1 + (-0.289 - 1.45i)T + (-0.923 + 0.382i)T^{2}
19 1+(−0.207−0.439i)T+(−0.634+0.773i)T2 1 + (-0.207 - 0.439i)T + (-0.634 + 0.773i)T^{2}
23 1+(0.520+0.427i)T+(0.195+0.980i)T2 1 + (0.520 + 0.427i)T + (0.195 + 0.980i)T^{2}
29 1+(−0.881+0.471i)T2 1 + (-0.881 + 0.471i)T^{2}
31 1+(−1.42−0.591i)T+(0.707+0.707i)T2 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2}
37 1+(0.773−0.634i)T2 1 + (0.773 - 0.634i)T^{2}
41 1+(−0.980+0.195i)T2 1 + (-0.980 + 0.195i)T^{2}
43 1+(−0.956+0.290i)T2 1 + (-0.956 + 0.290i)T^{2}
47 1+(−0.661+0.990i)T+(−0.382−0.923i)T2 1 + (-0.661 + 0.990i)T + (-0.382 - 0.923i)T^{2}
53 1+(−0.914−0.229i)T+(0.881+0.471i)T2 1 + (-0.914 - 0.229i)T + (0.881 + 0.471i)T^{2}
59 1+(0.0980−0.995i)T2 1 + (0.0980 - 0.995i)T^{2}
61 1+(0.612+0.825i)T+(−0.290+0.956i)T2 1 + (0.612 + 0.825i)T + (-0.290 + 0.956i)T^{2}
67 1+(−0.290+0.956i)T2 1 + (-0.290 + 0.956i)T^{2}
71 1+(0.831−0.555i)T2 1 + (0.831 - 0.555i)T^{2}
73 1+(−0.555+0.831i)T2 1 + (-0.555 + 0.831i)T^{2}
79 1+(−0.324+0.216i)T+(0.382−0.923i)T2 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2}
83 1+(−0.644+1.80i)T+(−0.773−0.634i)T2 1 + (-0.644 + 1.80i)T + (-0.773 - 0.634i)T^{2}
89 1+(−0.195+0.980i)T2 1 + (-0.195 + 0.980i)T^{2}
97 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)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

−8.578295210012615520072390020371, −7.79483356378443755599594605250, −7.51543328414920415679054480214, −6.50898773956875857532147537847, −6.12011491026010590008633843430, −5.11548742285944668706151601464, −3.94666967994144050975482344636, −2.94238316861300360392725392139, −1.88044472380026461251550002533, −0.857310613840104493683932138693, 0.71555231369157625272200241333, 2.50053718941235578314838234435, 3.10070752312427775977588122548, 3.97391045286354316691352943453, 4.62985694123827851750119067810, 5.64030384039491594543648555755, 6.76795646825276008721554448076, 7.49974021882900926702930034251, 8.105286206352819157397473625660, 8.767599841024583493700305401351

Graph of the ZZ-function along the critical line