↑

Properties

Label 2-304-1.1-c7-0-33
Degree 22
Conductor 304304
Sign −1-1
Analytic cond. 94.965094.9650
Root an. cond. 9.745009.74500
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 53.2·3-s + 88.7·5-s − 1.09e3·7-s + 645.·9-s − 247.·11-s + 5.31e3·13-s − 4.72e3·15-s + 8.38e3·17-s − 6.85e3·19-s + 5.82e4·21-s + 1.18e3·23-s − 7.02e4·25-s + 8.20e4·27-s + 1.63e5·29-s − 2.87e4·31-s + 1.31e4·33-s − 9.71e4·35-s + 2.10e5·37-s − 2.83e5·39-s + 6.36e5·41-s − 3.01e5·43-s + 5.73e4·45-s − 2.95e5·47-s + 3.73e5·49-s − 4.46e5·51-s + 4.64e5·53-s − 2.20e4·55-s + ⋯
L(s)  = 1  − 1.13·3-s + 0.317·5-s − 1.20·7-s + 0.295·9-s − 0.0561·11-s + 0.671·13-s − 0.361·15-s + 0.413·17-s − 0.229·19-s + 1.37·21-s + 0.0202·23-s − 0.899·25-s + 0.801·27-s + 1.24·29-s − 0.173·31-s + 0.0639·33-s − 0.382·35-s + 0.683·37-s − 0.764·39-s + 1.44·41-s − 0.578·43-s + 0.0938·45-s − 0.414·47-s + 0.453·49-s − 0.470·51-s + 0.428·53-s − 0.0178·55-s + ⋯

Functional equation

Λ(s)=(304s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(304s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 304304    =    24⋅192^{4} \cdot 19
Sign: −1-1
Analytic conductor: 94.965094.9650
Root analytic conductor: 9.745009.74500
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 304, ( :7/2), −1)(2,\ 304,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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
19 1+6.85e3T 1 + 6.85e3T
good3 1+53.2T+2.18e3T2 1 + 53.2T + 2.18e3T^{2}
5 1−88.7T+7.81e4T2 1 - 88.7T + 7.81e4T^{2}
7 1+1.09e3T+8.23e5T2 1 + 1.09e3T + 8.23e5T^{2}
11 1+247.T+1.94e7T2 1 + 247.T + 1.94e7T^{2}
13 1−5.31e3T+6.27e7T2 1 - 5.31e3T + 6.27e7T^{2}
17 1−8.38e3T+4.10e8T2 1 - 8.38e3T + 4.10e8T^{2}
23 1−1.18e3T+3.40e9T2 1 - 1.18e3T + 3.40e9T^{2}
29 1−1.63e5T+1.72e10T2 1 - 1.63e5T + 1.72e10T^{2}
31 1+2.87e4T+2.75e10T2 1 + 2.87e4T + 2.75e10T^{2}
37 1−2.10e5T+9.49e10T2 1 - 2.10e5T + 9.49e10T^{2}
41 1−6.36e5T+1.94e11T2 1 - 6.36e5T + 1.94e11T^{2}
43 1+3.01e5T+2.71e11T2 1 + 3.01e5T + 2.71e11T^{2}
47 1+2.95e5T+5.06e11T2 1 + 2.95e5T + 5.06e11T^{2}
53 1−4.64e5T+1.17e12T2 1 - 4.64e5T + 1.17e12T^{2}
59 1−1.03e6T+2.48e12T2 1 - 1.03e6T + 2.48e12T^{2}
61 1+1.16e6T+3.14e12T2 1 + 1.16e6T + 3.14e12T^{2}
67 1−2.70e6T+6.06e12T2 1 - 2.70e6T + 6.06e12T^{2}
71 1−3.21e6T+9.09e12T2 1 - 3.21e6T + 9.09e12T^{2}
73 1+5.62e5T+1.10e13T2 1 + 5.62e5T + 1.10e13T^{2}
79 1−1.08e6T+1.92e13T2 1 - 1.08e6T + 1.92e13T^{2}
83 1−1.24e6T+2.71e13T2 1 - 1.24e6T + 2.71e13T^{2}
89 1+3.49e6T+4.42e13T2 1 + 3.49e6T + 4.42e13T^{2}
97 1+2.42e6T+8.07e13T2 1 + 2.42e6T + 8.07e13T^{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

−10.13391782164916194107346074530, −9.331587963707626722105172962406, −8.098546769522139294562654760049, −6.68035477784931403085311810050, −6.14667553173674525951847749813, −5.29844916328595809263024542190, −3.94185780779164835239964158254, −2.70048375857876829386303417468, −1.03877455892235758701737169604, 0, 1.03877455892235758701737169604, 2.70048375857876829386303417468, 3.94185780779164835239964158254, 5.29844916328595809263024542190, 6.14667553173674525951847749813, 6.68035477784931403085311810050, 8.098546769522139294562654760049, 9.331587963707626722105172962406, 10.13391782164916194107346074530

Graph of the ZZ-function along the critical line