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Properties

Label 2-3549-273.116-c0-0-6
Degree 22
Conductor 35493549
Sign −0.301+0.953i-0.301 + 0.953i
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  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + 7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (−0.499 − 0.866i)16-s + (−1.5 + 0.866i)19-s + (0.5 − 0.866i)21-s + (0.5 − 0.866i)25-s − 0.999·27-s + (0.5 − 0.866i)28-s − 0.999·36-s + (1.5 − 0.866i)37-s − 43-s − 0.999·48-s + 49-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + 7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (−0.499 − 0.866i)16-s + (−1.5 + 0.866i)19-s + (0.5 − 0.866i)21-s + (0.5 − 0.866i)25-s − 0.999·27-s + (0.5 − 0.866i)28-s − 0.999·36-s + (1.5 − 0.866i)37-s − 43-s − 0.999·48-s + 49-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((−0.301+0.953i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((−0.301+0.953i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: −0.301+0.953i-0.301 + 0.953i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(2027,⋅)\chi_{3549} (2027, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), −0.301+0.953i)(2,\ 3549,\ (\ :0),\ -0.301 + 0.953i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7877131121.787713112
L(12)L(\frac12) ≈\approx 1.7877131121.787713112
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1−T 1 - T
13 1 1
good2 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
5 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}
79 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
97 1−1.73iT−T2 1 - 1.73iT - 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

−8.258345313121635897283444974163, −7.956596336802892185859258476163, −6.95075417051349831004792024554, −6.38289178660233816174991139505, −5.71184273714645165269111202039, −4.78435313950115788866885312619, −3.85003789344574746861109999485, −2.48642925924334335410721501670, −1.99002971842997088432551101453, −0.988385079620251236830817418952, 1.84470622892116988754578254410, 2.65549216054901037730773800983, 3.47547224503991014085091806831, 4.44098704077502661031754280930, 4.80937070239286446947957214667, 5.97051657210641814828685143893, 6.89935756452107476950234747110, 7.69223739874119623098489933960, 8.313887837299748150858049095623, 8.777046709203537479112421379895

Graph of the ZZ-function along the critical line