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Properties

Label 2-2368-1.1-c1-0-47
Degree 22
Conductor 23682368
Sign 11
Analytic cond. 18.908518.9085
Root an. cond. 4.348394.34839
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  + 2.14·3-s + 3.14·5-s + 1.74·7-s + 1.60·9-s + 2.89·11-s + 2.39·13-s + 6.74·15-s − 5.49·17-s − 2·19-s + 3.74·21-s − 1.60·23-s + 4.89·25-s − 3.00·27-s + 5.89·29-s + 3.94·31-s + 6.20·33-s + 5.49·35-s + 37-s + 5.14·39-s + 6.14·41-s − 5.20·43-s + 5.03·45-s + 0.253·47-s − 3.94·49-s − 11.7·51-s − 0.543·53-s + 9.09·55-s + ⋯
L(s)  = 1  + 1.23·3-s + 1.40·5-s + 0.660·7-s + 0.533·9-s + 0.871·11-s + 0.665·13-s + 1.74·15-s − 1.33·17-s − 0.458·19-s + 0.817·21-s − 0.333·23-s + 0.978·25-s − 0.577·27-s + 1.09·29-s + 0.708·31-s + 1.07·33-s + 0.928·35-s + 0.164·37-s + 0.823·39-s + 0.959·41-s − 0.793·43-s + 0.750·45-s + 0.0369·47-s − 0.564·49-s − 1.64·51-s − 0.0746·53-s + 1.22·55-s + ⋯

Functional equation

Λ(s)=(2368s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2368s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23682368    =    26⋅372^{6} \cdot 37
Sign: 11
Analytic conductor: 18.908518.9085
Root analytic conductor: 4.348394.34839
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2368, ( :1/2), 1)(2,\ 2368,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.9878632643.987863264
L(12)L(\frac12) ≈\approx 3.9878632643.987863264
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
37 1−T 1 - T
good3 1−2.14T+3T2 1 - 2.14T + 3T^{2}
5 1−3.14T+5T2 1 - 3.14T + 5T^{2}
7 1−1.74T+7T2 1 - 1.74T + 7T^{2}
11 1−2.89T+11T2 1 - 2.89T + 11T^{2}
13 1−2.39T+13T2 1 - 2.39T + 13T^{2}
17 1+5.49T+17T2 1 + 5.49T + 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 1+1.60T+23T2 1 + 1.60T + 23T^{2}
29 1−5.89T+29T2 1 - 5.89T + 29T^{2}
31 1−3.94T+31T2 1 - 3.94T + 31T^{2}
41 1−6.14T+41T2 1 - 6.14T + 41T^{2}
43 1+5.20T+43T2 1 + 5.20T + 43T^{2}
47 1−0.253T+47T2 1 - 0.253T + 47T^{2}
53 1+0.543T+53T2 1 + 0.543T + 53T^{2}
59 1+10.6T+59T2 1 + 10.6T + 59T^{2}
61 1+6.63T+61T2 1 + 6.63T + 61T^{2}
67 1+7.14T+67T2 1 + 7.14T + 67T^{2}
71 1+4.03T+71T2 1 + 4.03T + 71T^{2}
73 1−3.18T+73T2 1 - 3.18T + 73T^{2}
79 1−9.89T+79T2 1 - 9.89T + 79T^{2}
83 1+6.03T+83T2 1 + 6.03T + 83T^{2}
89 1−4.50T+89T2 1 - 4.50T + 89T^{2}
97 1−12.9T+97T2 1 - 12.9T + 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

−8.915447509081912887975860156796, −8.485462029047928033319831977019, −7.63126703147705746927336782213, −6.41735797637935806682001338552, −6.16796841201063004777512128194, −4.86793964355662857915740254160, −4.11338539453528498659353500163, −2.97415100636859777003739925392, −2.13652759010808021764488466576, −1.45163804074301669692462615906, 1.45163804074301669692462615906, 2.13652759010808021764488466576, 2.97415100636859777003739925392, 4.11338539453528498659353500163, 4.86793964355662857915740254160, 6.16796841201063004777512128194, 6.41735797637935806682001338552, 7.63126703147705746927336782213, 8.485462029047928033319831977019, 8.915447509081912887975860156796

Graph of the ZZ-function along the critical line