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Properties

Label 4-2695275-1.1-c1e2-0-30
Degree 44
Conductor 26952752695275
Sign −1-1
Analytic cond. 171.853171.853
Root an. cond. 3.620673.62067
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  − 4-s − 11-s − 3·16-s + 25-s − 14·31-s + 16·37-s + 44-s − 6·47-s − 4·49-s − 6·53-s + 12·59-s + 7·64-s − 2·67-s + 6·71-s − 18·89-s + 4·97-s − 100-s + 16·103-s + 12·113-s + 121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.301·11-s − 3/4·16-s + 1/5·25-s − 2.51·31-s + 2.63·37-s + 0.150·44-s − 0.875·47-s − 4/7·49-s − 0.824·53-s + 1.56·59-s + 7/8·64-s − 0.244·67-s + 0.712·71-s − 1.90·89-s + 0.406·97-s − 0.0999·100-s + 1.57·103-s + 1.12·113-s + 1/11·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(2695275s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2695275s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 26952752695275    =    34⋅52⋅1133^{4} \cdot 5^{2} \cdot 11^{3}
Sign: −1-1
Analytic conductor: 171.853171.853
Root analytic conductor: 3.620673.62067
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 2695275, ( :1/2,1/2), −1)(4,\ 2695275,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
11C1C_1 1+T 1 + T
good2C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
7C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
19C22C_2^2 1+28T2+p2T4 1 + 28 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2×\timesC2C_2 (1+4T+pT2)(1+10T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
43C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−6T+pT2)(1+12T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2×\timesC2C_2 (1+pT2)(1+6T+pT2) ( 1 + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−12T+pT2)(1+pT2) ( 1 - 12 T + p T^{2} )( 1 + p T^{2} )
61C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−2T+pT2)(1+4T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
73C22C_2^2 1−122T2+p2T4 1 - 122 T^{2} + p^{2} T^{4}
79C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
83C22C_2^2 1−62T2+p2T4 1 - 62 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+6T+pT2)(1+12T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−14T+pT2)(1+10T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 10 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

−7.36828250900503417275095834972, −7.08427307801830795641955571937, −6.56770762327839215421996821253, −6.10544281123673679328718070451, −5.68752923116124310684381104274, −5.28337990826132128961816141009, −4.74340941157951287670358044241, −4.45701172528653151946818565300, −3.91174842651292112836066981197, −3.44997528715722504141228418169, −2.89616505606515762031794926470, −2.25423524075303761594945182227, −1.80090193517497396536391751852, −0.870024250421061824697470711606, 0, 0.870024250421061824697470711606, 1.80090193517497396536391751852, 2.25423524075303761594945182227, 2.89616505606515762031794926470, 3.44997528715722504141228418169, 3.91174842651292112836066981197, 4.45701172528653151946818565300, 4.74340941157951287670358044241, 5.28337990826132128961816141009, 5.68752923116124310684381104274, 6.10544281123673679328718070451, 6.56770762327839215421996821253, 7.08427307801830795641955571937, 7.36828250900503417275095834972

Graph of the ZZ-function along the critical line