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Properties

Label 2-616-1.1-c3-0-39
Degree 22
Conductor 616616
Sign −1-1
Analytic cond. 36.345136.3451
Root an. cond. 6.028696.02869
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  − 0.761·3-s + 15.6·5-s + 7·7-s − 26.4·9-s − 11·11-s − 0.574·13-s − 11.9·15-s − 129.·17-s − 35.3·19-s − 5.33·21-s − 182.·23-s + 119.·25-s + 40.6·27-s − 41.5·29-s + 96.4·31-s + 8.37·33-s + 109.·35-s − 172.·37-s + 0.437·39-s − 385.·41-s + 360.·43-s − 413.·45-s + 72.5·47-s + 49·49-s + 98.8·51-s + 153.·53-s − 172.·55-s + ⋯
L(s)  = 1  − 0.146·3-s + 1.39·5-s + 0.377·7-s − 0.978·9-s − 0.301·11-s − 0.0122·13-s − 0.205·15-s − 1.85·17-s − 0.427·19-s − 0.0554·21-s − 1.65·23-s + 0.957·25-s + 0.290·27-s − 0.266·29-s + 0.558·31-s + 0.0441·33-s + 0.528·35-s − 0.766·37-s + 0.00179·39-s − 1.46·41-s + 1.28·43-s − 1.36·45-s + 0.225·47-s + 0.142·49-s + 0.271·51-s + 0.397·53-s − 0.421·55-s + ⋯

Functional equation

Λ(s)=(616s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(616s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    23⋅7⋅112^{3} \cdot 7 \cdot 11
Sign: −1-1
Analytic conductor: 36.345136.3451
Root analytic conductor: 6.028696.02869
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 616, ( :3/2), −1)(2,\ 616,\ (\ :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
7 1−7T 1 - 7T
11 1+11T 1 + 11T
good3 1+0.761T+27T2 1 + 0.761T + 27T^{2}
5 1−15.6T+125T2 1 - 15.6T + 125T^{2}
13 1+0.574T+2.19e3T2 1 + 0.574T + 2.19e3T^{2}
17 1+129.T+4.91e3T2 1 + 129.T + 4.91e3T^{2}
19 1+35.3T+6.85e3T2 1 + 35.3T + 6.85e3T^{2}
23 1+182.T+1.21e4T2 1 + 182.T + 1.21e4T^{2}
29 1+41.5T+2.43e4T2 1 + 41.5T + 2.43e4T^{2}
31 1−96.4T+2.97e4T2 1 - 96.4T + 2.97e4T^{2}
37 1+172.T+5.06e4T2 1 + 172.T + 5.06e4T^{2}
41 1+385.T+6.89e4T2 1 + 385.T + 6.89e4T^{2}
43 1−360.T+7.95e4T2 1 - 360.T + 7.95e4T^{2}
47 1−72.5T+1.03e5T2 1 - 72.5T + 1.03e5T^{2}
53 1−153.T+1.48e5T2 1 - 153.T + 1.48e5T^{2}
59 1−327.T+2.05e5T2 1 - 327.T + 2.05e5T^{2}
61 1−42.4T+2.26e5T2 1 - 42.4T + 2.26e5T^{2}
67 1−572.T+3.00e5T2 1 - 572.T + 3.00e5T^{2}
71 1+784.T+3.57e5T2 1 + 784.T + 3.57e5T^{2}
73 1+452.T+3.89e5T2 1 + 452.T + 3.89e5T^{2}
79 1+557.T+4.93e5T2 1 + 557.T + 4.93e5T^{2}
83 1+160.T+5.71e5T2 1 + 160.T + 5.71e5T^{2}
89 1+395.T+7.04e5T2 1 + 395.T + 7.04e5T^{2}
97 1+1.09e3T+9.12e5T2 1 + 1.09e3T + 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

−9.858459557290438366342213137677, −8.866051262012561106834287146330, −8.294968706009296684933199447217, −6.87420981841409295357834622395, −6.04046047924669748548694766044, −5.37925943780929913833081469135, −4.24473359222431907117372524547, −2.58561068702553925785434231230, −1.86142104909081744572740522094, 0, 1.86142104909081744572740522094, 2.58561068702553925785434231230, 4.24473359222431907117372524547, 5.37925943780929913833081469135, 6.04046047924669748548694766044, 6.87420981841409295357834622395, 8.294968706009296684933199447217, 8.866051262012561106834287146330, 9.858459557290438366342213137677

Graph of the ZZ-function along the critical line