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Properties

Label 2-320892-1.1-c1-0-7
Degree 22
Conductor 320892320892
Sign 11
Analytic cond. 2562.332562.33
Root an. cond. 50.619550.6195
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-s + 2·5-s + 4·7-s + 9-s + 13-s − 2·15-s − 17-s − 2·19-s − 4·21-s − 8·23-s − 25-s − 27-s − 2·29-s + 4·31-s + 8·35-s + 8·37-s − 39-s + 10·41-s + 2·45-s − 8·47-s + 9·49-s + 51-s − 2·53-s + 2·57-s − 6·61-s + 4·63-s + 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 1.35·35-s + 1.31·37-s − 0.160·39-s + 1.56·41-s + 0.298·45-s − 1.16·47-s + 9/7·49-s + 0.140·51-s − 0.274·53-s + 0.264·57-s − 0.768·61-s + 0.503·63-s + 0.248·65-s + ⋯

Functional equation

Λ(s)=(320892s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(320892s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320892320892    =    22⋅3⋅112⋅13⋅172^{2} \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 2562.332562.33
Root analytic conductor: 50.619550.6195
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 320892, ( :1/2), 1)(2,\ 320892,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2728691063.272869106
L(12)L(\frac12) ≈\approx 3.2728691063.272869106
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+T 1 + T
11 1 1
13 1−T 1 - T
17 1+T 1 + T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−16T+pT2 1 - 16 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+4T+pT2 1 + 4 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

−12.42567500405820, −12.20916278058390, −11.66636121973856, −11.16464377841798, −10.89112882809236, −10.49111553334423, −9.874151839579496, −9.445418602038917, −9.146500656091921, −8.257852618813157, −7.933647444872601, −7.834734486897635, −6.930549367335189, −6.394213047228979, −5.990681962399778, −5.688618404479301, −5.040368685538649, −4.599704487899022, −4.196761749936330, −3.647713759106965, −2.704088108385258, −2.076814269665488, −1.861657159233084, −1.172973284902923, −0.5069787985010591, 0.5069787985010591, 1.172973284902923, 1.861657159233084, 2.076814269665488, 2.704088108385258, 3.647713759106965, 4.196761749936330, 4.599704487899022, 5.040368685538649, 5.688618404479301, 5.990681962399778, 6.394213047228979, 6.930549367335189, 7.834734486897635, 7.933647444872601, 8.257852618813157, 9.146500656091921, 9.445418602038917, 9.874151839579496, 10.49111553334423, 10.89112882809236, 11.16464377841798, 11.66636121973856, 12.20916278058390, 12.42567500405820

Graph of the ZZ-function along the critical line