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Properties

Label 2-396-1.1-c3-0-12
Degree 22
Conductor 396396
Sign −1-1
Analytic cond. 23.364723.3647
Root an. cond. 4.833714.83371
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17.1·5-s − 23.1·7-s − 11·11-s − 59.6·13-s − 80.8·17-s − 11.7·19-s − 56.0·23-s + 168.·25-s − 85.4·29-s − 99.8·31-s − 396.·35-s + 402.·37-s + 27.0·41-s − 74·43-s − 408.·47-s + 192.·49-s + 463.·53-s − 188.·55-s − 498.·59-s − 635.·61-s − 1.02e3·65-s − 701.·67-s + 27.7·71-s + 619.·73-s + 254.·77-s − 208.·79-s − 1.30e3·83-s + ⋯
L(s)  = 1  + 1.53·5-s − 1.24·7-s − 0.301·11-s − 1.27·13-s − 1.15·17-s − 0.141·19-s − 0.508·23-s + 1.34·25-s − 0.547·29-s − 0.578·31-s − 1.91·35-s + 1.79·37-s + 0.103·41-s − 0.262·43-s − 1.26·47-s + 0.560·49-s + 1.20·53-s − 0.462·55-s − 1.10·59-s − 1.33·61-s − 1.95·65-s − 1.27·67-s + 0.0463·71-s + 0.993·73-s + 0.376·77-s − 0.296·79-s − 1.72·83-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: −1-1
Analytic conductor: 23.364723.3647
Root analytic conductor: 4.833714.83371
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 396, ( :3/2), −1)(2,\ 396,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
11 1+11T 1 + 11T
good5 1−17.1T+125T2 1 - 17.1T + 125T^{2}
7 1+23.1T+343T2 1 + 23.1T + 343T^{2}
13 1+59.6T+2.19e3T2 1 + 59.6T + 2.19e3T^{2}
17 1+80.8T+4.91e3T2 1 + 80.8T + 4.91e3T^{2}
19 1+11.7T+6.85e3T2 1 + 11.7T + 6.85e3T^{2}
23 1+56.0T+1.21e4T2 1 + 56.0T + 1.21e4T^{2}
29 1+85.4T+2.43e4T2 1 + 85.4T + 2.43e4T^{2}
31 1+99.8T+2.97e4T2 1 + 99.8T + 2.97e4T^{2}
37 1−402.T+5.06e4T2 1 - 402.T + 5.06e4T^{2}
41 1−27.0T+6.89e4T2 1 - 27.0T + 6.89e4T^{2}
43 1+74T+7.95e4T2 1 + 74T + 7.95e4T^{2}
47 1+408.T+1.03e5T2 1 + 408.T + 1.03e5T^{2}
53 1−463.T+1.48e5T2 1 - 463.T + 1.48e5T^{2}
59 1+498.T+2.05e5T2 1 + 498.T + 2.05e5T^{2}
61 1+635.T+2.26e5T2 1 + 635.T + 2.26e5T^{2}
67 1+701.T+3.00e5T2 1 + 701.T + 3.00e5T^{2}
71 1−27.7T+3.57e5T2 1 - 27.7T + 3.57e5T^{2}
73 1−619.T+3.89e5T2 1 - 619.T + 3.89e5T^{2}
79 1+208.T+4.93e5T2 1 + 208.T + 4.93e5T^{2}
83 1+1.30e3T+5.71e5T2 1 + 1.30e3T + 5.71e5T^{2}
89 1+1.39e3T+7.04e5T2 1 + 1.39e3T + 7.04e5T^{2}
97 1−1.05e3T+9.12e5T2 1 - 1.05e3T + 9.12e5T^{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

−10.05783308804332371999961816570, −9.691943699057387933092912451342, −8.904757943762688612686988469143, −7.41064726322153901475281195952, −6.42913890955334638661628131567, −5.76781803290502380869367139818, −4.56896407020150818853070963648, −2.91448017618907003083281129995, −2.01237317689166792041705731153, 0, 2.01237317689166792041705731153, 2.91448017618907003083281129995, 4.56896407020150818853070963648, 5.76781803290502380869367139818, 6.42913890955334638661628131567, 7.41064726322153901475281195952, 8.904757943762688612686988469143, 9.691943699057387933092912451342, 10.05783308804332371999961816570

Graph of the ZZ-function along the critical line