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Properties

Label 4-130e2-1.1-c1e2-0-9
Degree 44
Conductor 1690016900
Sign 11
Analytic cond. 1.077551.07755
Root an. cond. 1.018841.01884
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  + 4·3-s − 4-s + 6·9-s − 4·12-s + 4·13-s + 16-s − 12·17-s − 25-s − 4·27-s + 12·29-s − 6·36-s + 16·39-s − 20·43-s + 4·48-s + 14·49-s − 48·51-s − 4·52-s + 20·61-s − 64-s + 12·68-s − 4·75-s − 16·79-s − 37·81-s + 48·87-s + 100-s + 12·101-s − 8·103-s + ⋯
L(s)  = 1  + 2.30·3-s − 1/2·4-s + 2·9-s − 1.15·12-s + 1.10·13-s + 1/4·16-s − 2.91·17-s − 1/5·25-s − 0.769·27-s + 2.22·29-s − 36-s + 2.56·39-s − 3.04·43-s + 0.577·48-s + 2·49-s − 6.72·51-s − 0.554·52-s + 2.56·61-s − 1/8·64-s + 1.45·68-s − 0.461·75-s − 1.80·79-s − 4.11·81-s + 5.14·87-s + 1/10·100-s + 1.19·101-s − 0.788·103-s + ⋯

Functional equation

Λ(s)=(16900s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(16900s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1690016900    =    22⋅52⋅1322^{2} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 1.077551.07755
Root analytic conductor: 1.018841.01884
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 16900, ( :1/2,1/2), 1)(4,\ 16900,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8818630921.881863092
L(12)L(\frac12) ≈\approx 1.8818630921.881863092
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)
bad2C2C_2 1+T2 1 + T^{2}
5C2C_2 1+T2 1 + T^{2}
13C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
7C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
37C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
61C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
67C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
97C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 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

−13.64307047223434482753417733184, −13.43987587270077914606363645205, −12.97672677219211980506742795396, −12.15278791750272641970502876374, −11.34035261918977020073503627788, −11.16198642625201982590408812968, −10.12234744829263090851293025623, −9.892917837024798160963779878530, −8.925613416825172513155250554034, −8.798807137075199390827783279547, −8.406467371345004101137487683045, −8.225449997921241172760293684223, −7.12850970394112456487857799337, −6.69135004082542710953257671703, −5.87667466683688625617498021739, −4.77795400527083453366709640303, −4.12831713785439225613911682394, −3.49913214415705381439925680590, −2.70495821028726688122985006903, −2.02481506755937410809719002151, 2.02481506755937410809719002151, 2.70495821028726688122985006903, 3.49913214415705381439925680590, 4.12831713785439225613911682394, 4.77795400527083453366709640303, 5.87667466683688625617498021739, 6.69135004082542710953257671703, 7.12850970394112456487857799337, 8.225449997921241172760293684223, 8.406467371345004101137487683045, 8.798807137075199390827783279547, 8.925613416825172513155250554034, 9.892917837024798160963779878530, 10.12234744829263090851293025623, 11.16198642625201982590408812968, 11.34035261918977020073503627788, 12.15278791750272641970502876374, 12.97672677219211980506742795396, 13.43987587270077914606363645205, 13.64307047223434482753417733184

Graph of the ZZ-function along the critical line