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Properties

Label 2-1920-240.149-c0-0-3
Degree 22
Conductor 19201920
Sign −0.382+0.923i-0.382 + 0.923i
Analytic cond. 0.9582040.958204
Root an. cond. 0.9788790.978879
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  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s − 1.00i·9-s − 1.00·15-s − 1.41·17-s + (1 − i)19-s − 1.41i·23-s + 1.00i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 + 1.00i)51-s − 1.41i·57-s + (−1 + i)61-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s − 1.00i·9-s − 1.00·15-s − 1.41·17-s + (1 − i)19-s − 1.41i·23-s + 1.00i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 + 1.00i)51-s − 1.41i·57-s + (−1 + i)61-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s) L(s)=((−0.382+0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: −0.382+0.923i-0.382 + 0.923i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1920(1889,⋅)\chi_{1920} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :0), −0.382+0.923i)(2,\ 1920,\ (\ :0),\ -0.382 + 0.923i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1528845201.152884520
L(12)L(\frac12) ≈\approx 1.1528845201.152884520
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)
bad2 1 1
3 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good7 1+T2 1 + T^{2}
11 1−iT2 1 - iT^{2}
13 1+iT2 1 + iT^{2}
17 1+1.41T+T2 1 + 1.41T + T^{2}
19 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
23 1+1.41iT−T2 1 + 1.41iT - T^{2}
29 1+iT2 1 + iT^{2}
31 1+T2 1 + T^{2}
37 1−iT2 1 - iT^{2}
41 1+T2 1 + T^{2}
43 1−iT2 1 - iT^{2}
47 1−1.41T+T2 1 - 1.41T + T^{2}
53 1+iT2 1 + iT^{2}
59 1−iT2 1 - iT^{2}
61 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1−2T+T2 1 - 2T + T^{2}
83 1−iT2 1 - iT^{2}
89 1+T2 1 + 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

−8.897388278363709874948411656893, −8.498316364981478303411875626758, −7.58309357102454823863004241707, −6.99616676341377960169103458985, −6.15482831036092311680169594509, −4.89964597872888503742949498674, −4.21035481134347670454651788897, −3.13451694512454482195420610889, −2.17186569299943386037840278972, −0.77884833184188450067356919233, 1.95778654842782115418464481243, 3.08555575098576558348423473765, 3.72322816787976880019680435729, 4.54147855639560133824732523319, 5.53044713783399615094760808943, 6.60406978827796684763351198882, 7.55273875562359617614173084054, 7.962387441856983550635079094090, 8.942714261574991466509451618012, 9.571572762585456116195893508332

Graph of the ZZ-function along the critical line