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Properties

Label 2-4304-1.1-c1-0-126
Degree 22
Conductor 43044304
Sign −1-1
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.01·3-s + 0.217·5-s + 2.74·7-s − 1.97·9-s − 1.10·11-s + 3.99·13-s + 0.219·15-s − 2.81·17-s − 8.43·19-s + 2.78·21-s − 1.44·23-s − 4.95·25-s − 5.03·27-s − 5.00·29-s − 3.45·31-s − 1.11·33-s + 0.597·35-s + 2.96·37-s + 4.04·39-s − 11.5·41-s − 5.07·43-s − 0.429·45-s + 8.95·47-s + 0.555·49-s − 2.84·51-s − 2.67·53-s − 0.240·55-s + ⋯
L(s)  = 1  + 0.584·3-s + 0.0971·5-s + 1.03·7-s − 0.658·9-s − 0.333·11-s + 1.10·13-s + 0.0567·15-s − 0.683·17-s − 1.93·19-s + 0.606·21-s − 0.301·23-s − 0.990·25-s − 0.968·27-s − 0.928·29-s − 0.620·31-s − 0.194·33-s + 0.100·35-s + 0.487·37-s + 0.647·39-s − 1.80·41-s − 0.774·43-s − 0.0640·45-s + 1.30·47-s + 0.0793·49-s − 0.399·51-s − 0.367·53-s − 0.0324·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: −1-1
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4304, ( :1/2), −1)(2,\ 4304,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
269 1+T 1 + T
good3 1−1.01T+3T2 1 - 1.01T + 3T^{2}
5 1−0.217T+5T2 1 - 0.217T + 5T^{2}
7 1−2.74T+7T2 1 - 2.74T + 7T^{2}
11 1+1.10T+11T2 1 + 1.10T + 11T^{2}
13 1−3.99T+13T2 1 - 3.99T + 13T^{2}
17 1+2.81T+17T2 1 + 2.81T + 17T^{2}
19 1+8.43T+19T2 1 + 8.43T + 19T^{2}
23 1+1.44T+23T2 1 + 1.44T + 23T^{2}
29 1+5.00T+29T2 1 + 5.00T + 29T^{2}
31 1+3.45T+31T2 1 + 3.45T + 31T^{2}
37 1−2.96T+37T2 1 - 2.96T + 37T^{2}
41 1+11.5T+41T2 1 + 11.5T + 41T^{2}
43 1+5.07T+43T2 1 + 5.07T + 43T^{2}
47 1−8.95T+47T2 1 - 8.95T + 47T^{2}
53 1+2.67T+53T2 1 + 2.67T + 53T^{2}
59 1−3.44T+59T2 1 - 3.44T + 59T^{2}
61 1+11.5T+61T2 1 + 11.5T + 61T^{2}
67 1−4.43T+67T2 1 - 4.43T + 67T^{2}
71 1−3.84T+71T2 1 - 3.84T + 71T^{2}
73 1+10.9T+73T2 1 + 10.9T + 73T^{2}
79 1−5.55T+79T2 1 - 5.55T + 79T^{2}
83 1−4.47T+83T2 1 - 4.47T + 83T^{2}
89 1−10.8T+89T2 1 - 10.8T + 89T^{2}
97 1+2.92T+97T2 1 + 2.92T + 97T^{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

−8.155221864994629413201702068004, −7.54811321498126270013033008106, −6.43445697474110199744977298543, −5.88378507623541834164081025135, −4.99738876303499431009940249758, −4.12682303081906551045817828786, −3.45141509478160475883082190090, −2.21614335610604889972346402159, −1.77499255526640895610597875253, 0, 1.77499255526640895610597875253, 2.21614335610604889972346402159, 3.45141509478160475883082190090, 4.12682303081906551045817828786, 4.99738876303499431009940249758, 5.88378507623541834164081025135, 6.43445697474110199744977298543, 7.54811321498126270013033008106, 8.155221864994629413201702068004

Graph of the ZZ-function along the critical line