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Properties

Label 2-3872-968.115-c0-0-0
Degree 22
Conductor 38723872
Sign 0.704−0.709i0.704 - 0.709i
Analytic cond. 1.932371.93237
Root an. cond. 1.390101.39010
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  + (1.25 + 0.910i)3-s + (0.431 + 1.32i)9-s + (−0.0855 − 0.996i)11-s + (1.74 + 0.859i)17-s + (0.0798 − 0.151i)19-s + (0.696 − 0.717i)25-s + (−0.190 + 0.584i)27-s + (0.799 − 1.32i)33-s + (−1.38 − 0.495i)41-s + (−1.29 + 1.48i)43-s + (0.516 + 0.856i)49-s + (1.40 + 2.66i)51-s + (0.237 − 0.116i)57-s + (0.683 − 0.243i)59-s + (−1.55 + 0.455i)67-s + ⋯
L(s)  = 1  + (1.25 + 0.910i)3-s + (0.431 + 1.32i)9-s + (−0.0855 − 0.996i)11-s + (1.74 + 0.859i)17-s + (0.0798 − 0.151i)19-s + (0.696 − 0.717i)25-s + (−0.190 + 0.584i)27-s + (0.799 − 1.32i)33-s + (−1.38 − 0.495i)41-s + (−1.29 + 1.48i)43-s + (0.516 + 0.856i)49-s + (1.40 + 2.66i)51-s + (0.237 − 0.116i)57-s + (0.683 − 0.243i)59-s + (−1.55 + 0.455i)67-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((0.704−0.709i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s) L(s)=((0.704−0.709i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: 0.704−0.709i0.704 - 0.709i
Analytic conductor: 1.932371.93237
Root analytic conductor: 1.390101.39010
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3872(3503,⋅)\chi_{3872} (3503, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :0), 0.704−0.709i)(2,\ 3872,\ (\ :0),\ 0.704 - 0.709i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.0988281712.098828171
L(12)L(\frac12) ≈\approx 2.0988281712.098828171
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 1
11 1+(0.0855+0.996i)T 1 + (0.0855 + 0.996i)T
good3 1+(−1.25−0.910i)T+(0.309+0.951i)T2 1 + (-1.25 - 0.910i)T + (0.309 + 0.951i)T^{2}
5 1+(−0.696+0.717i)T2 1 + (-0.696 + 0.717i)T^{2}
7 1+(−0.516−0.856i)T2 1 + (-0.516 - 0.856i)T^{2}
13 1+(−0.198−0.980i)T2 1 + (-0.198 - 0.980i)T^{2}
17 1+(−1.74−0.859i)T+(0.610+0.791i)T2 1 + (-1.74 - 0.859i)T + (0.610 + 0.791i)T^{2}
19 1+(−0.0798+0.151i)T+(−0.564−0.825i)T2 1 + (-0.0798 + 0.151i)T + (-0.564 - 0.825i)T^{2}
23 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
29 1+(0.0285−0.999i)T2 1 + (0.0285 - 0.999i)T^{2}
31 1+(0.870+0.491i)T2 1 + (0.870 + 0.491i)T^{2}
37 1+(−0.993+0.113i)T2 1 + (-0.993 + 0.113i)T^{2}
41 1+(1.38+0.495i)T+(0.774+0.633i)T2 1 + (1.38 + 0.495i)T + (0.774 + 0.633i)T^{2}
43 1+(1.29−1.48i)T+(−0.142−0.989i)T2 1 + (1.29 - 1.48i)T + (-0.142 - 0.989i)T^{2}
47 1+(0.254−0.967i)T2 1 + (0.254 - 0.967i)T^{2}
53 1+(−0.0855+0.996i)T2 1 + (-0.0855 + 0.996i)T^{2}
59 1+(−0.683+0.243i)T+(0.774−0.633i)T2 1 + (-0.683 + 0.243i)T + (0.774 - 0.633i)T^{2}
61 1+(0.362+0.931i)T2 1 + (0.362 + 0.931i)T^{2}
67 1+(1.55−0.455i)T+(0.841−0.540i)T2 1 + (1.55 - 0.455i)T + (0.841 - 0.540i)T^{2}
71 1+(−0.941−0.336i)T2 1 + (-0.941 - 0.336i)T^{2}
73 1+(0.100+0.498i)T+(−0.921+0.389i)T2 1 + (0.100 + 0.498i)T + (-0.921 + 0.389i)T^{2}
79 1+(0.466+0.884i)T2 1 + (0.466 + 0.884i)T^{2}
83 1+(0.614−0.0704i)T+(0.974−0.226i)T2 1 + (0.614 - 0.0704i)T + (0.974 - 0.226i)T^{2}
89 1+(0.282−1.96i)T+(−0.959−0.281i)T2 1 + (0.282 - 1.96i)T + (-0.959 - 0.281i)T^{2}
97 1+(−0.0526−0.0222i)T+(0.696+0.717i)T2 1 + (-0.0526 - 0.0222i)T + (0.696 + 0.717i)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

−8.632307090768507216523028012127, −8.250193143162000944546242283681, −7.62965792862559665537273852143, −6.51797143304972866582115791978, −5.67301802582601766918792535891, −4.86001941190505896675027565306, −3.93713965444812598830810390589, −3.28387578298978949497134580313, −2.74382725622103481946497863980, −1.38641621338907185013499016185, 1.26504912898755318526177097343, 2.07163742079041253326576193788, 3.05464037221820705874749948846, 3.57462880689048540547153251559, 4.83955171355713846229992763088, 5.54297516892521503831729061705, 6.75966884236119482349628280699, 7.25972541848680278769392872455, 7.72135410211519533402015845136, 8.530928427437506503868224493222

Graph of the ZZ-function along the critical line