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Properties

Label 2-15e3-75.56-c0-0-0
Degree 22
Conductor 33753375
Sign 0.770−0.637i0.770 - 0.637i
Analytic cond. 1.684341.68434
Root an. cond. 1.297821.29782
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  + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s − 0.618·7-s + (−0.587 + 0.809i)8-s + (0.951 − 0.309i)11-s + (−0.363 + 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.5 − 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.309 − 0.224i)28-s + (0.951 + 1.30i)29-s − i·32-s + (0.190 − 0.587i)34-s + (−0.5 + 1.53i)37-s + (0.587 + 0.190i)38-s + ⋯
L(s)  = 1  + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s − 0.618·7-s + (−0.587 + 0.809i)8-s + (0.951 − 0.309i)11-s + (−0.363 + 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (0.5 − 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.309 − 0.224i)28-s + (0.951 + 1.30i)29-s − i·32-s + (0.190 − 0.587i)34-s + (−0.5 + 1.53i)37-s + (0.587 + 0.190i)38-s + ⋯

Functional equation

Λ(s)=(3375s/2 ΓC(s) L(s)=((0.770−0.637i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3375s/2 ΓC(s) L(s)=((0.770−0.637i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33753375    =    33⋅533^{3} \cdot 5^{3}
Sign: 0.770−0.637i0.770 - 0.637i
Analytic conductor: 1.684341.68434
Root analytic conductor: 1.297821.29782
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3375(26,⋅)\chi_{3375} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3375, ( :0), 0.770−0.637i)(2,\ 3375,\ (\ :0),\ 0.770 - 0.637i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3807153291.380715329
L(12)L(\frac12) ≈\approx 1.3807153291.380715329
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)
bad3 1 1
5 1 1
good2 1+(−0.587+0.190i)T+(0.809−0.587i)T2 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2}
7 1+0.618T+T2 1 + 0.618T + T^{2}
11 1+(−0.951+0.309i)T+(0.809−0.587i)T2 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}
13 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
17 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
19 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
23 1+(0.951−0.309i)T+(0.809−0.587i)T2 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2}
29 1+(−0.951−1.30i)T+(−0.309+0.951i)T2 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2}
31 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
37 1+(0.5−1.53i)T+(−0.809−0.587i)T2 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}
41 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
43 1+T2 1 + T^{2}
47 1+(−0.951−1.30i)T+(−0.309+0.951i)T2 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2}
53 1+(−0.363−0.5i)T+(−0.309+0.951i)T2 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2}
59 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
61 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
67 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
71 1+(−0.951−1.30i)T+(−0.309+0.951i)T2 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2}
73 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
79 1+(1.30−0.951i)T+(0.309−0.951i)T2 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}
83 1+(−0.363+0.5i)T+(−0.309−0.951i)T2 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2}
89 1+(−0.951+0.309i)T+(0.809−0.587i)T2 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}
97 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)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

−8.914709093613294634087650922584, −8.177951826718702642677008181575, −7.41148149961304476251936588104, −6.49580441405776329845077131596, −5.76970952376119644712926847724, −4.99746651758078235522363372333, −4.14045904344825707025925792716, −3.36247672170627758410762225312, −2.82570393338442623337829796625, −1.23797899058232323698002500727, 0.810738413735601281582788373788, 2.23231532146518522368893779632, 3.55458023567640894416274516656, 3.98559631915088052707009646969, 4.87235673141337868489788547633, 5.80243054361561462943397091587, 6.25858317941318452388920783453, 7.02526505541967281770161235613, 7.926915384663545089319119093808, 8.897361276246088711709662470295

Graph of the ZZ-function along the critical line