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Properties

Label 2-3549-21.11-c0-0-4
Degree 22
Conductor 35493549
Sign 0.997+0.0633i0.997 + 0.0633i
Analytic cond. 1.771181.77118
Root an. cond. 1.330851.33085
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)2-s + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s − 0.999·6-s + (0.5 + 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + (0.866 + 0.499i)14-s − 0.999·15-s + (0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.499i)18-s − 0.999i·21-s + (0.866 − 0.5i)23-s + (0.500 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s − 0.999·6-s + (0.5 + 0.866i)7-s + i·8-s + (0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + (0.866 + 0.499i)14-s − 0.999·15-s + (0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.499i)18-s − 0.999i·21-s + (0.866 − 0.5i)23-s + (0.500 − 0.866i)24-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((0.997+0.0633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((0.997+0.0633i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: 0.997+0.0633i0.997 + 0.0633i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(1691,⋅)\chi_{3549} (1691, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), 0.997+0.0633i)(2,\ 3549,\ (\ :0),\ 0.997 + 0.0633i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7456558441.745655844
L(12)L(\frac12) ≈\approx 1.7456558441.745655844
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1 1
good2 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
5 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
19 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
23 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
29 1−iT−T2 1 - iT - T^{2}
31 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1−iT−T2 1 - iT - T^{2}
43 1−T+T2 1 - T + T^{2}
47 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
53 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
59 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+iT−T2 1 + iT - T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
97 1−T+T2 1 - T + 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

−8.867118623682439267179453959178, −8.006901599910303953208350087610, −7.08702646211805955442728500833, −6.17714019574574606602296400339, −5.54347531447502642399740918064, −4.91216716290624993071109739580, −4.53438768286729147973005780310, −3.05918275912218564190744286965, −2.22168550145716425123315655499, −1.44005862046710614988005490001, 0.944096726641429801040237829216, 2.32622147804246965531194554793, 3.86824556852502356809292992330, 4.15644397995543290419976810357, 5.08736872898104167330005748207, 5.75498117445487873371770651125, 6.25736525739070097178661276442, 6.99529117170494004340601762642, 7.58711698941694326817981893621, 8.981654196578344938069948324397

Graph of the ZZ-function along the critical line