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Properties

Label 2-3675-21.2-c0-0-2
Degree 22
Conductor 36753675
Sign 0.126−0.991i0.126 - 0.991i
Analytic cond. 1.834061.83406
Root an. cond. 1.354271.35427
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)3-s + (−0.5 − 0.866i)4-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)16-s + (−1.73 + i)17-s + 0.999i·27-s − 0.999·36-s + (1.73 + i)47-s − 0.999i·48-s + (0.999 − 1.73i)51-s + 0.999·64-s + (1.73 + 0.999i)68-s + (−1 + 1.73i)79-s + (−0.5 − 0.866i)81-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)16-s + (−1.73 + i)17-s + 0.999i·27-s − 0.999·36-s + (1.73 + i)47-s − 0.999i·48-s + (0.999 − 1.73i)51-s + 0.999·64-s + (1.73 + 0.999i)68-s + (−1 + 1.73i)79-s + (−0.5 − 0.866i)81-s + ⋯

Functional equation

Λ(s)=(3675s/2 ΓC(s) L(s)=((0.126−0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3675s/2 ΓC(s) L(s)=((0.126−0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36753675    =    3⋅52⋅723 \cdot 5^{2} \cdot 7^{2}
Sign: 0.126−0.991i0.126 - 0.991i
Analytic conductor: 1.834061.83406
Root analytic conductor: 1.354271.35427
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3675(1451,⋅)\chi_{3675} (1451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3675, ( :0), 0.126−0.991i)(2,\ 3675,\ (\ :0),\ 0.126 - 0.991i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50578834870.5057883487
L(12)L(\frac12) ≈\approx 0.50578834870.5057883487
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
5 1 1
7 1 1
good2 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}
13 1+T2 1 + T^{2}
17 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
19 1+(−0.5−0.866i)T2 1 + (-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+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1+(−1.73−i)T+(0.5+0.866i)T2 1 + (-1.73 - i)T + (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)T2 1 + (-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+(−0.5+0.866i)T2 1 + (-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 1−2iT−T2 1 - 2iT - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1+T2 1 + 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

−9.014305596728288564997868550168, −8.400282079088037896568499781843, −7.15992871073387845969347451222, −6.40488010956900813157396670089, −5.90451412257362294042994342512, −5.11463551762738628865296784253, −4.36988783741455222517272017635, −3.85837619821449973749196039083, −2.30439102216675380266862881058, −1.11557228854828374353590489623, 0.37646360222908505327163146353, 2.01247329896486790040110599672, 2.92387641823813717208499251324, 4.15967593893003488904474067750, 4.67969709517985621770026115620, 5.49483563171297673495130064646, 6.43902700047500532244408601960, 7.15416042612149016755022297401, 7.58511405441797821968322754305, 8.608612589234536758683088139332

Graph of the ZZ-function along the critical line