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Properties

Label 2-39e2-13.7-c0-0-1
Degree 22
Conductor 15211521
Sign 0.674−0.738i0.674 - 0.738i
Analytic cond. 0.7590770.759077
Root an. cond. 0.8712500.871250
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  + (−0.866 + 0.5i)4-s + (1.36 + 0.366i)7-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)19-s − i·25-s + (−1.36 + 0.366i)28-s + (1 + i)31-s + (0.366 + 1.36i)37-s + (0.866 + 0.5i)49-s + 0.999i·64-s + (1.36 − 0.366i)67-s + (−1 + i)73-s + (−0.366 − 1.36i)76-s + (0.366 − 1.36i)97-s + (0.5 + 0.866i)100-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)4-s + (1.36 + 0.366i)7-s + (0.499 − 0.866i)16-s + (−0.366 + 1.36i)19-s − i·25-s + (−1.36 + 0.366i)28-s + (1 + i)31-s + (0.366 + 1.36i)37-s + (0.866 + 0.5i)49-s + 0.999i·64-s + (1.36 − 0.366i)67-s + (−1 + i)73-s + (−0.366 − 1.36i)76-s + (0.366 − 1.36i)97-s + (0.5 + 0.866i)100-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=((0.674−0.738i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s) L(s)=((0.674−0.738i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: 0.674−0.738i0.674 - 0.738i
Analytic conductor: 0.7590770.759077
Root analytic conductor: 0.8712500.871250
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1521(1333,⋅)\chi_{1521} (1333, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1521, ( :0), 0.674−0.738i)(2,\ 1521,\ (\ :0),\ 0.674 - 0.738i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0345868891.034586889
L(12)L(\frac12) ≈\approx 1.0345868891.034586889
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
13 1 1
good2 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
5 1+iT2 1 + iT^{2}
7 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
11 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
23 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
31 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
37 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
41 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
43 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
47 1−iT2 1 - iT^{2}
53 1+T2 1 + T^{2}
59 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
71 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
73 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
79 1+T2 1 + T^{2}
83 1+iT2 1 + iT^{2}
89 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
97 1+(−0.366+1.36i)T+(−0.866−0.5i)T2 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)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

−9.789983730860939749090604346396, −8.635401092113366327746898034725, −8.309615159805725908289861330521, −7.71047864039964680507588133831, −6.50127339193516017243931254201, −5.44727529413500105361017666323, −4.71105790625125623305125133738, −4.01381583352518620004423396348, −2.80054720559142132140513939114, −1.46383854567660906090243976228, 1.00590992146060072376580203962, 2.27257178374952788949614248738, 3.83953811371073736117525611212, 4.64762352851899894365460135792, 5.19795534094704571975896521050, 6.17477414005074460699207809921, 7.30603228816637395553731706194, 8.027026064507285190245537838033, 8.827211079583198201941644455174, 9.431850757414943487974547202953

Graph of the ZZ-function along the critical line