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Properties

Label 2-3872-88.75-c0-0-1
Degree 22
Conductor 38723872
Sign 0.0219−0.999i0.0219 - 0.999i
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  + (−1.30 + 0.951i)3-s + (0.500 − 1.53i)9-s + (0.190 + 0.587i)17-s + (0.5 − 0.363i)19-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)41-s + 1.61·43-s + (0.309 + 0.951i)49-s + (−0.809 − 0.587i)51-s + (−0.309 + 0.951i)57-s + (0.5 + 0.363i)59-s − 0.618·67-s + (1.30 + 0.951i)73-s + (0.5 − 1.53i)75-s + ⋯
L(s)  = 1  + (−1.30 + 0.951i)3-s + (0.500 − 1.53i)9-s + (0.190 + 0.587i)17-s + (0.5 − 0.363i)19-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (1.30 − 0.951i)41-s + 1.61·43-s + (0.309 + 0.951i)49-s + (−0.809 − 0.587i)51-s + (−0.309 + 0.951i)57-s + (0.5 + 0.363i)59-s − 0.618·67-s + (1.30 + 0.951i)73-s + (0.5 − 1.53i)75-s + ⋯

Functional equation

Λ(s)=(3872s/2 ΓC(s) L(s)=((0.0219−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3872s/2 ΓC(s) L(s)=((0.0219−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38723872    =    25⋅1122^{5} \cdot 11^{2}
Sign: 0.0219−0.999i0.0219 - 0.999i
Analytic conductor: 1.932371.93237
Root analytic conductor: 1.390101.39010
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3872(2671,⋅)\chi_{3872} (2671, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3872, ( :0), 0.0219−0.999i)(2,\ 3872,\ (\ :0),\ 0.0219 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73839083070.7383908307
L(12)L(\frac12) ≈\approx 0.73839083070.7383908307
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 1
good3 1+(1.30−0.951i)T+(0.309−0.951i)T2 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}
5 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
7 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
13 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
17 1+(−0.190−0.587i)T+(−0.809+0.587i)T2 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}
19 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2}
23 1−T2 1 - T^{2}
29 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
31 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
37 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
41 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}
43 1−1.61T+T2 1 - 1.61T + T^{2}
47 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
53 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
59 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
61 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
67 1+0.618T+T2 1 + 0.618T + T^{2}
71 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
73 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
79 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
83 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
89 1+1.61T+T2 1 + 1.61T + T^{2}
97 1+(0.5−1.53i)T+(−0.809−0.587i)T2 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)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

−9.196409396962537377056976679090, −8.051718782102147034269799119664, −7.28235121253237444136835847260, −6.40456257275428403565794729717, −5.65869692664908462425563826728, −5.29530044953628164443853671495, −4.23443281559646161965441633025, −3.82776802643958777810612639163, −2.53757958295007691175271558561, −1.02671392762518929445375979857, 0.63160312063844911773506738522, 1.70270477977171873244679121242, 2.76391064844183213849282200554, 4.03246293479021643975295774981, 4.92602962460192068984586526441, 5.72133682082126376360895262484, 6.14097983279626670588289387284, 7.00615430887541538212191911658, 7.55480749976502482445946363062, 8.202288729785028763356844294480

Graph of the ZZ-function along the critical line