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Properties

Label 4-117e2-1.1-c1e2-0-5
Degree 44
Conductor 1368913689
Sign 11
Analytic cond. 0.8728220.872822
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 9·7-s − 5·13-s − 6·19-s + 10·25-s − 18·28-s − 12·37-s + 13·43-s + 47·49-s + 10·52-s − 13·61-s + 8·64-s − 21·67-s + 12·76-s + 26·79-s − 45·91-s − 33·97-s − 20·100-s − 26·103-s − 11·121-s + 127-s + 131-s − 54·133-s + 137-s + 139-s + 24·148-s + 149-s + ⋯
L(s)  = 1  − 4-s + 3.40·7-s − 1.38·13-s − 1.37·19-s + 2·25-s − 3.40·28-s − 1.97·37-s + 1.98·43-s + 47/7·49-s + 1.38·52-s − 1.66·61-s + 64-s − 2.56·67-s + 1.37·76-s + 2.92·79-s − 4.71·91-s − 3.35·97-s − 2·100-s − 2.56·103-s − 121-s + 0.0887·127-s + 0.0873·131-s − 4.68·133-s + 0.0854·137-s + 0.0848·139-s + 1.97·148-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(13689s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(13689s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1368913689    =    34⋅1323^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 0.8728220.872822
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 13689, ( :1/2,1/2), 1)(4,\ 13689,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1044107671.104410767
L(12)L(\frac12) ≈\approx 1.1044107671.104410767
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
13C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good2C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C2C_2 (1−5T+pT2)(1−4T+pT2) ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} )
11C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C2C_2 (1−T+pT2)(1+7T+pT2) ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} )
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
31C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C2C_2 (1+T+pT2)(1+11T+pT2) ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} )
41C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)(1−5T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} )
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
61C2C_2 (1−T+pT2)(1+14T+pT2) ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} )
67C2C_2 (1+5T+pT2)(1+16T+pT2) ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
73C2C_2 (1−17T+pT2)(1+17T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C2C_2 (1−13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}
83C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
89C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
97C2C_2 (1+14T+pT2)(1+19T+pT2) ( 1 + 14 T + p T^{2} )( 1 + 19 T + p T^{2} )
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.81279173160257393401615679087, −13.62694379516947677042840371391, −12.46427648873793078139292504378, −12.37128003121127970556062776597, −11.80718845771292300073660906113, −11.01560413161178701888136302047, −10.69394454445487923535021362642, −10.50640332215871967047634143278, −9.162552205587415278270033255433, −9.067439377155730272574142174468, −8.328349868159648652891769166710, −8.047159372711070323146310603239, −7.41249293760598380020186165924, −6.78510443580637797647023510823, −5.43648123126854160264107828925, −5.10354134532696297849656395253, −4.44536589169225904437065558856, −4.35334440313498098980548492336, −2.51377898048347983367857045794, −1.56257140765442249388862088029, 1.56257140765442249388862088029, 2.51377898048347983367857045794, 4.35334440313498098980548492336, 4.44536589169225904437065558856, 5.10354134532696297849656395253, 5.43648123126854160264107828925, 6.78510443580637797647023510823, 7.41249293760598380020186165924, 8.047159372711070323146310603239, 8.328349868159648652891769166710, 9.067439377155730272574142174468, 9.162552205587415278270033255433, 10.50640332215871967047634143278, 10.69394454445487923535021362642, 11.01560413161178701888136302047, 11.80718845771292300073660906113, 12.37128003121127970556062776597, 12.46427648873793078139292504378, 13.62694379516947677042840371391, 13.81279173160257393401615679087

Graph of the ZZ-function along the critical line