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Properties

Label 2-39e2-13.7-c0-0-0
Degree 22
Conductor 15211521
Sign 0.999+0.0386i0.999 + 0.0386i
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  + (−1.36 + 0.366i)2-s + (0.866 − 0.5i)4-s + (−1 − i)5-s + (1.73 + i)10-s + (0.366 + 1.36i)11-s + (−0.499 + 0.866i)16-s + (−1.36 − 0.366i)20-s + (−1 − 1.73i)22-s + i·25-s + (0.366 − 1.36i)32-s + (1.36 − 0.366i)41-s + (1.73 − i)43-s + (1 + 0.999i)44-s + (1 − i)47-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)50-s + ⋯
L(s)  = 1  + (−1.36 + 0.366i)2-s + (0.866 − 0.5i)4-s + (−1 − i)5-s + (1.73 + i)10-s + (0.366 + 1.36i)11-s + (−0.499 + 0.866i)16-s + (−1.36 − 0.366i)20-s + (−1 − 1.73i)22-s + i·25-s + (0.366 − 1.36i)32-s + (1.36 − 0.366i)41-s + (1.73 − i)43-s + (1 + 0.999i)44-s + (1 − i)47-s + (−0.866 − 0.5i)49-s + (−0.366 − 1.36i)50-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=((0.999+0.0386i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s) L(s)=((0.999+0.0386i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: 0.999+0.0386i0.999 + 0.0386i
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), 0.999+0.0386i)(2,\ 1521,\ (\ :0),\ 0.999 + 0.0386i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.44138682280.4413868228
L(12)L(\frac12) ≈\approx 0.44138682280.4413868228
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+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2}
5 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
7 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
11 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(−0.866−0.5i)T2 1 + (-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+iT2 1 + iT^{2}
37 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
41 1+(−1.36+0.366i)T+(0.866−0.5i)T2 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2}
43 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
47 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
53 1+T2 1 + T^{2}
59 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
71 1+(0.366−1.36i)T+(−0.866−0.5i)T2 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2}
73 1−iT2 1 - iT^{2}
79 1+T2 1 + T^{2}
83 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
89 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
97 1+(−0.866−0.5i)T2 1 + (-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

−9.433887188630957730582729891218, −8.806668342244214125187169839931, −8.195991541986127494232348791197, −7.36161599127269745674835161000, −6.99422507541652918191920193203, −5.65189500314977577782628096744, −4.47262492310976724208332708394, −3.96832653364552590580840849863, −2.10151424966125618268324803474, −0.824396872911944168976828545014, 0.910976812824809821657976240477, 2.54243219611971349729838532073, 3.35961610706580435650370450950, 4.39267900832395775358801435636, 5.84994662107991887093620573936, 6.71747595334367771324026584177, 7.69403531890582378433062542675, 7.985696200793325042684205832657, 8.972028883293363817145065015291, 9.517621805985242699266950205914

Graph of the ZZ-function along the critical line