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Properties

Label 4-3520e2-1.1-c0e2-0-1
Degree 44
Conductor 1239040012390400
Sign 11
Analytic cond. 3.086023.08602
Root an. cond. 1.325401.32540
Motivic weight 00
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  − 2·5-s − 2·9-s + 3·25-s − 4·37-s + 4·45-s − 2·49-s − 4·53-s + 3·81-s − 4·89-s − 121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 8·185-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·5-s − 2·9-s + 3·25-s − 4·37-s + 4·45-s − 2·49-s − 4·53-s + 3·81-s − 4·89-s − 121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 8·185-s + 191-s + 193-s + ⋯

Functional equation

Λ(s)=(12390400s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(12390400s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1239040012390400    =    212⋅52⋅1122^{12} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 3.086023.08602
Root analytic conductor: 1.325401.32540
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 12390400, ( :0,0), 1)(4,\ 12390400,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.090499493920.09049949392
L(12)L(\frac12) ≈\approx 0.090499493920.09049949392
L(1)L(1) 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)
bad2 1 1
5C1C_1 (1+T)2 ( 1 + T )^{2}
11C2C_2 1+T2 1 + T^{2}
good3C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
7C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
13C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
17C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
19C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
23C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
29C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
37C1C_1 (1+T)4 ( 1 + T )^{4}
41C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
43C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
47C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
53C1C_1 (1+T)4 ( 1 + T )^{4}
59C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
79C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C1C_1 (1+T)4 ( 1 + T )^{4}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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

−8.906272173887560652249424208287, −8.391125004428543911437387827701, −8.239777849152239474998307888799, −7.904789609333678077107079491194, −7.70361026860171457198786144363, −6.89215499809786808363089351634, −6.88052814560047899578070815707, −6.51000948881895579438659990939, −5.93019594615811831594958349597, −5.36136573580389731364830629085, −5.22799277169442149717506573343, −4.71755390172079802577866346334, −4.38005243966857215626882863804, −3.76688908177301172060046679210, −3.33632972757570360585191736771, −3.04826805153953325300783283596, −2.96693692365797606635619518738, −1.94035255978083368585762446169, −1.42428804254149585396632902344, −0.17542455797828592980790210424, 0.17542455797828592980790210424, 1.42428804254149585396632902344, 1.94035255978083368585762446169, 2.96693692365797606635619518738, 3.04826805153953325300783283596, 3.33632972757570360585191736771, 3.76688908177301172060046679210, 4.38005243966857215626882863804, 4.71755390172079802577866346334, 5.22799277169442149717506573343, 5.36136573580389731364830629085, 5.93019594615811831594958349597, 6.51000948881895579438659990939, 6.88052814560047899578070815707, 6.89215499809786808363089351634, 7.70361026860171457198786144363, 7.904789609333678077107079491194, 8.239777849152239474998307888799, 8.391125004428543911437387827701, 8.906272173887560652249424208287

Graph of the ZZ-function along the critical line