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Properties

Label 2-29988-1.1-c1-0-12
Degree 22
Conductor 2998829988
Sign 11
Analytic cond. 239.455239.455
Root an. cond. 15.474315.4743
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 4·11-s − 13-s + 17-s + 4·23-s + 4·25-s + 29-s + 3·31-s + 2·37-s + 5·41-s + 6·43-s − 9·47-s + 2·53-s − 12·55-s − 59-s − 12·61-s + 3·65-s + 10·67-s + 8·73-s + 5·83-s − 3·85-s + 6·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.20·11-s − 0.277·13-s + 0.242·17-s + 0.834·23-s + 4/5·25-s + 0.185·29-s + 0.538·31-s + 0.328·37-s + 0.780·41-s + 0.914·43-s − 1.31·47-s + 0.274·53-s − 1.61·55-s − 0.130·59-s − 1.53·61-s + 0.372·65-s + 1.22·67-s + 0.936·73-s + 0.548·83-s − 0.325·85-s + 0.635·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

Λ(s)=(29988s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29988s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2998829988    =    22⋅32⋅72⋅172^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 239.455239.455
Root analytic conductor: 15.474315.4743
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 29988, ( :1/2), 1)(2,\ 29988,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7876308791.787630879
L(12)L(\frac12) ≈\approx 1.7876308791.787630879
L(32)L(\frac{3}{2}) 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 1
7 1 1
17 1−T 1 - T
good5 1+3T+pT2 1 + 3 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+T+pT2 1 + T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−5T+pT2 1 - 5 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−12T+pT2 1 - 12 T + p 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

−15.00383546613591, −14.75695485800430, −14.22135295162351, −13.58233535452926, −12.91638272193689, −12.24785765184672, −12.05590430922745, −11.39440221388327, −11.03329771384316, −10.42968573263102, −9.489685624958141, −9.312621958378128, −8.491916333462923, −8.017339105936953, −7.486602661894247, −6.888226284864712, −6.391913820257550, −5.648414135331871, −4.729205921573479, −4.402278047976074, −3.648861089865249, −3.217342986380655, −2.322882840105712, −1.262939526443419, −0.5743808226760613, 0.5743808226760613, 1.262939526443419, 2.322882840105712, 3.217342986380655, 3.648861089865249, 4.402278047976074, 4.729205921573479, 5.648414135331871, 6.391913820257550, 6.888226284864712, 7.486602661894247, 8.017339105936953, 8.491916333462923, 9.312621958378128, 9.489685624958141, 10.42968573263102, 11.03329771384316, 11.39440221388327, 12.05590430922745, 12.24785765184672, 12.91638272193689, 13.58233535452926, 14.22135295162351, 14.75695485800430, 15.00383546613591

Graph of the ZZ-function along the critical line