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Properties

Label 4-495e2-1.1-c2e2-0-1
Degree 44
Conductor 245025245025
Sign 11
Analytic cond. 181.920181.920
Root an. cond. 3.672573.67257
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 10·5-s + 22·11-s − 7·16-s + 30·20-s + 75·25-s + 36·31-s − 66·44-s − 78·49-s − 220·55-s + 204·59-s + 69·64-s + 156·71-s + 70·80-s − 4·89-s − 225·100-s + 363·121-s − 108·124-s − 500·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 360·155-s + 157-s + ⋯
L(s)  = 1  − 3/4·4-s − 2·5-s + 2·11-s − 0.437·16-s + 3/2·20-s + 3·25-s + 1.16·31-s − 3/2·44-s − 1.59·49-s − 4·55-s + 3.45·59-s + 1.07·64-s + 2.19·71-s + 7/8·80-s − 0.0449·89-s − 9/4·100-s + 3·121-s − 0.870·124-s − 4·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 2.32·155-s + 0.00636·157-s + ⋯

Functional equation

Λ(s)=(245025s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(245025s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 245025245025    =    34⋅52⋅1123^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 181.920181.920
Root analytic conductor: 3.672573.67257
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 245025, ( :1,1), 1)(4,\ 245025,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.1844355351.184435535
L(12)L(\frac12) ≈\approx 1.1844355351.184435535
L(2)L(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 (1+pT)2 ( 1 + p T )^{2}
11C1C_1 (1−pT)2 ( 1 - p T )^{2}
good2C22C_2^2 1+3T2+p4T4 1 + 3 T^{2} + p^{4} T^{4}
7C22C_2^2 1+78T2+p4T4 1 + 78 T^{2} + p^{4} T^{4}
13C22C_2^2 1−162T2+p4T4 1 - 162 T^{2} + p^{4} T^{4}
17C22C_2^2 1−402T2+p4T4 1 - 402 T^{2} + p^{4} T^{4}
19C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
23C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
29C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
31C2C_2 (1−18T+p2T2)2 ( 1 - 18 T + p^{2} T^{2} )^{2}
37C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
41C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
43C22C_2^2 1−3522T2+p4T4 1 - 3522 T^{2} + p^{4} T^{4}
47C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
53C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
59C2C_2 (1−102T+p2T2)2 ( 1 - 102 T + p^{2} T^{2} )^{2}
61C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
67C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
71C2C_2 (1−78T+p2T2)2 ( 1 - 78 T + p^{2} T^{2} )^{2}
73C22C_2^2 1+10638T2+p4T4 1 + 10638 T^{2} + p^{4} T^{4}
79C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
83C22C_2^2 1−13602T2+p4T4 1 - 13602 T^{2} + p^{4} T^{4}
89C2C_2 (1+2T+p2T2)2 ( 1 + 2 T + p^{2} T^{2} )^{2}
97C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + 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

−11.25946579749753796905904202857, −10.61414615836775017677238830271, −9.931119631226211048762280334786, −9.580161549977617725825212824531, −9.084163100437264029272454974741, −8.662474599469984518573554130797, −8.159345742428135514450714262116, −8.152450871173124757084201175585, −7.20968898667823856517125185813, −6.84795390293104492097642295673, −6.63300945157482431724516193468, −5.88659291936216063069131394204, −4.88280687063990011902003605870, −4.78709964565121954169614645242, −4.00857876958298732717803296327, −3.83898473998158960902208696932, −3.34477470609688061775631436523, −2.38696157012541458771915640738, −1.16057423029661443238809608699, −0.53589133734196648335068086781, 0.53589133734196648335068086781, 1.16057423029661443238809608699, 2.38696157012541458771915640738, 3.34477470609688061775631436523, 3.83898473998158960902208696932, 4.00857876958298732717803296327, 4.78709964565121954169614645242, 4.88280687063990011902003605870, 5.88659291936216063069131394204, 6.63300945157482431724516193468, 6.84795390293104492097642295673, 7.20968898667823856517125185813, 8.152450871173124757084201175585, 8.159345742428135514450714262116, 8.662474599469984518573554130797, 9.084163100437264029272454974741, 9.580161549977617725825212824531, 9.931119631226211048762280334786, 10.61414615836775017677238830271, 11.25946579749753796905904202857

Graph of the ZZ-function along the critical line