↑

Properties

Label 2-7360-1.1-c1-0-5
Degree 22
Conductor 73607360
Sign 11
Analytic cond. 58.769858.7698
Root an. cond. 7.666157.66615
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·3-s − 5-s − 1.56·7-s + 3.56·9-s − 2·11-s − 0.561·13-s + 2.56·15-s − 1.56·17-s − 6·19-s + 4·21-s + 23-s + 25-s − 1.43·27-s + 2.12·29-s − 9.24·31-s + 5.12·33-s + 1.56·35-s + 0.438·37-s + 1.43·39-s − 4.12·41-s − 3.56·45-s − 7.68·47-s − 4.56·49-s + 4·51-s + 0.438·53-s + 2·55-s + 15.3·57-s + ⋯
L(s)  = 1  − 1.47·3-s − 0.447·5-s − 0.590·7-s + 1.18·9-s − 0.603·11-s − 0.155·13-s + 0.661·15-s − 0.378·17-s − 1.37·19-s + 0.872·21-s + 0.208·23-s + 0.200·25-s − 0.276·27-s + 0.394·29-s − 1.66·31-s + 0.891·33-s + 0.263·35-s + 0.0720·37-s + 0.230·39-s − 0.643·41-s − 0.530·45-s − 1.12·47-s − 0.651·49-s + 0.560·51-s + 0.0602·53-s + 0.269·55-s + 2.03·57-s + ⋯

Functional equation

Λ(s)=(7360s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7360s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 73607360    =    26⋅5⋅232^{6} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 58.769858.7698
Root analytic conductor: 7.666157.66615
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7360, ( :1/2), 1)(2,\ 7360,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.12899678990.1289967899
L(12)L(\frac12) ≈\approx 0.12899678990.1289967899
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
23 1−T 1 - T
good3 1+2.56T+3T2 1 + 2.56T + 3T^{2}
7 1+1.56T+7T2 1 + 1.56T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+0.561T+13T2 1 + 0.561T + 13T^{2}
17 1+1.56T+17T2 1 + 1.56T + 17T^{2}
19 1+6T+19T2 1 + 6T + 19T^{2}
29 1−2.12T+29T2 1 - 2.12T + 29T^{2}
31 1+9.24T+31T2 1 + 9.24T + 31T^{2}
37 1−0.438T+37T2 1 - 0.438T + 37T^{2}
41 1+4.12T+41T2 1 + 4.12T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+7.68T+47T2 1 + 7.68T + 47T^{2}
53 1−0.438T+53T2 1 - 0.438T + 53T^{2}
59 1+8.68T+59T2 1 + 8.68T + 59T^{2}
61 1+1.12T+61T2 1 + 1.12T + 61T^{2}
67 1−4.43T+67T2 1 - 4.43T + 67T^{2}
71 1−1.87T+71T2 1 - 1.87T + 71T^{2}
73 1+8.56T+73T2 1 + 8.56T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1+14.9T+83T2 1 + 14.9T + 83T^{2}
89 1+2.24T+89T2 1 + 2.24T + 89T^{2}
97 1+4.87T+97T2 1 + 4.87T + 97T^{2}
show more
show less
   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

−7.80337201638930396026904987793, −6.90705818757530641793191366558, −6.55186184829324675613080525884, −5.82523467346732724361261848658, −5.12464647385414978552669869920, −4.53166996611470291734280220608, −3.70876848149490520475873350230, −2.72982507247659226570294411801, −1.59020388430224735192338928215, −0.19297313723472051299983867567, 0.19297313723472051299983867567, 1.59020388430224735192338928215, 2.72982507247659226570294411801, 3.70876848149490520475873350230, 4.53166996611470291734280220608, 5.12464647385414978552669869920, 5.82523467346732724361261848658, 6.55186184829324675613080525884, 6.90705818757530641793191366558, 7.80337201638930396026904987793

Graph of the ZZ-function along the critical line