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Properties

Label 2-320-1.1-c1-0-5
Degree 22
Conductor 320320
Sign 11
Analytic cond. 2.555212.55521
Root an. cond. 1.598501.59850
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s − 5-s + 2.82·7-s + 5.00·9-s − 5.65·11-s + 2·13-s − 2.82·15-s + 2·17-s + 8.00·21-s − 2.82·23-s + 25-s + 5.65·27-s − 6·29-s − 5.65·31-s − 16.0·33-s − 2.82·35-s + 10·37-s + 5.65·39-s + 2·41-s − 8.48·43-s − 5.00·45-s + 2.82·47-s + 1.00·49-s + 5.65·51-s − 6·53-s + 5.65·55-s + 11.3·59-s + ⋯
L(s)  = 1  + 1.63·3-s − 0.447·5-s + 1.06·7-s + 1.66·9-s − 1.70·11-s + 0.554·13-s − 0.730·15-s + 0.485·17-s + 1.74·21-s − 0.589·23-s + 0.200·25-s + 1.08·27-s − 1.11·29-s − 1.01·31-s − 2.78·33-s − 0.478·35-s + 1.64·37-s + 0.905·39-s + 0.312·41-s − 1.29·43-s − 0.745·45-s + 0.412·47-s + 0.142·49-s + 0.792·51-s − 0.824·53-s + 0.762·55-s + 1.47·59-s + ⋯

Functional equation

Λ(s)=(320s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(320s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320320    =    26⋅52^{6} \cdot 5
Sign: 11
Analytic conductor: 2.555212.55521
Root analytic conductor: 1.598501.59850
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 320, ( :1/2), 1)(2,\ 320,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0928453982.092845398
L(12)L(\frac12) ≈\approx 2.0928453982.092845398
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
5 1+T 1 + T
good3 1−2.82T+3T2 1 - 2.82T + 3T^{2}
7 1−2.82T+7T2 1 - 2.82T + 7T^{2}
11 1+5.65T+11T2 1 + 5.65T + 11T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+2.82T+23T2 1 + 2.82T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1+5.65T+31T2 1 + 5.65T + 31T^{2}
37 1−10T+37T2 1 - 10T + 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+8.48T+43T2 1 + 8.48T + 43T^{2}
47 1−2.82T+47T2 1 - 2.82T + 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1−11.3T+59T2 1 - 11.3T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+2.82T+67T2 1 + 2.82T + 67T^{2}
71 1+5.65T+71T2 1 + 5.65T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+11.3T+79T2 1 + 11.3T + 79T^{2}
83 1−2.82T+83T2 1 - 2.82T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−11.53817667191451373632568628568, −10.63652798535804361642446776952, −9.626869907218388433642260600245, −8.519721980105615848195832680186, −7.943670102081038157537312274234, −7.42867144191638870823373674443, −5.51245569181223824742894392200, −4.27226961306955603738878899232, −3.12925247737475439352160521810, −1.95006760437674357452119030431, 1.95006760437674357452119030431, 3.12925247737475439352160521810, 4.27226961306955603738878899232, 5.51245569181223824742894392200, 7.42867144191638870823373674443, 7.943670102081038157537312274234, 8.519721980105615848195832680186, 9.626869907218388433642260600245, 10.63652798535804361642446776952, 11.53817667191451373632568628568

Graph of the ZZ-function along the critical line