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Properties

Label 2-1740-1.1-c1-0-19
Degree 22
Conductor 17401740
Sign −1-1
Analytic cond. 13.893913.8939
Root an. cond. 3.727463.72746
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 3·7-s + 9-s − 3·11-s + 13-s + 15-s − 3·17-s − 6·19-s − 3·21-s − 4·23-s + 25-s + 27-s + 29-s − 4·31-s − 3·33-s − 3·35-s − 4·37-s + 39-s + 2·41-s + 4·43-s + 45-s − 3·47-s + 2·49-s − 3·51-s + 6·53-s − 3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s − 1.37·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.185·29-s − 0.718·31-s − 0.522·33-s − 0.507·35-s − 0.657·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.437·47-s + 2/7·49-s − 0.420·51-s + 0.824·53-s − 0.404·55-s + ⋯

Functional equation

Λ(s)=(1740s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1740s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17401740    =    22⋅3⋅5⋅292^{2} \cdot 3 \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 13.893913.8939
Root analytic conductor: 3.727463.72746
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1740, ( :1/2), −1)(2,\ 1740,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
3 1−T 1 - T
5 1−T 1 - T
29 1−T 1 - T
good7 1+3T+pT2 1 + 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1+9T+pT2 1 + 9 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−7T+pT2 1 - 7 T + p T^{2}
97 1+14T+pT2 1 + 14 T + p T^{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

−8.977772545976198663458639648741, −8.248029066882421296740697658734, −7.33548636345869317829741092214, −6.45094562380598595733597865632, −5.87968209883743333563161571072, −4.68670697707933784822821040344, −3.73949324550628996073299112294, −2.77551981441276545461339322850, −1.94725580514987939336132871333, 0, 1.94725580514987939336132871333, 2.77551981441276545461339322850, 3.73949324550628996073299112294, 4.68670697707933784822821040344, 5.87968209883743333563161571072, 6.45094562380598595733597865632, 7.33548636345869317829741092214, 8.248029066882421296740697658734, 8.977772545976198663458639648741

Graph of the ZZ-function along the critical line