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Properties

Label 4-3240e2-1.1-c1e2-0-29
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 669.336669.336
Root an. cond. 5.086405.08640
Motivic weight 11
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  + 2·5-s + 7-s + 5·11-s + 5·13-s + 4·17-s + 2·19-s − 7·23-s + 3·25-s + 5·29-s + 5·31-s + 2·35-s − 6·37-s + 8·43-s + 5·47-s + 49-s + 9·53-s + 10·55-s − 26·59-s + 2·61-s + 10·65-s − 14·67-s − 9·71-s + 5·77-s + 6·79-s + 10·83-s + 8·85-s + 3·89-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s + 1.50·11-s + 1.38·13-s + 0.970·17-s + 0.458·19-s − 1.45·23-s + 3/5·25-s + 0.928·29-s + 0.898·31-s + 0.338·35-s − 0.986·37-s + 1.21·43-s + 0.729·47-s + 1/7·49-s + 1.23·53-s + 1.34·55-s − 3.38·59-s + 0.256·61-s + 1.24·65-s − 1.71·67-s − 1.06·71-s + 0.569·77-s + 0.675·79-s + 1.09·83-s + 0.867·85-s + 0.317·89-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 669.336669.336
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :1/2,1/2), 1)(4,\ 10497600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.3566612425.356661242
L(12)L(\frac12) ≈\approx 5.3566612425.356661242
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)
bad2 1 1
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
good7D4D_{4} 1−T−pT3+p2T4 1 - T - p T^{3} + p^{2} T^{4}
11C22C_2^2 1−5T+14T2−5pT3+p2T4 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4}
13D4D_{4} 1−5T+18T2−5pT3+p2T4 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
23D4D_{4} 1+7T+44T2+7pT3+p2T4 1 + 7 T + 44 T^{2} + 7 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−5T+50T2−5pT3+p2T4 1 - 5 T + 50 T^{2} - 5 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−5T+54T2−5pT3+p2T4 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+6T+26T2+6pT3+p2T4 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+25T2+p2T4 1 + 25 T^{2} + p^{2} T^{4}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47D4D_{4} 1−5T+86T2−5pT3+p2T4 1 - 5 T + 86 T^{2} - 5 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−9T+112T2−9pT3+p2T4 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4}
59C2C_2 (1+13T+pT2)2 ( 1 + 13 T + p T^{2} )^{2}
61D4D_{4} 1−2T+66T2−2pT3+p2T4 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+14T+126T2+14pT3+p2T4 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+9T+148T2+9pT3+p2T4 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4}
73C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
79D4D_{4} 1−6T+110T2−6pT3+p2T4 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−10T+134T2−10pT3+p2T4 1 - 10 T + 134 T^{2} - 10 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−3T+52T2−3pT3+p2T4 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4}
97C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{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

−8.857449078695155521324121511884, −8.720762709195757263662235507376, −7.917225906548906442004130191206, −7.87589259388261715397818721514, −7.40472637544268798116812868346, −6.93137968145181140621046171731, −6.35870535421854895138558621178, −6.17766391193468089129121537353, −5.93421836585419308708142466618, −5.66778610808101394820397488795, −4.85053597454650241862841687672, −4.73688837916018369756851686823, −4.06095534746289918622858142520, −3.84430790532800964179544769989, −3.16432154335307708493304060967, −3.02314884188694069630644257720, −2.03295347094569248608226426374, −1.82764502667012712630972120616, −1.13902324336306079739609225599, −0.844460583937210432502758867826, 0.844460583937210432502758867826, 1.13902324336306079739609225599, 1.82764502667012712630972120616, 2.03295347094569248608226426374, 3.02314884188694069630644257720, 3.16432154335307708493304060967, 3.84430790532800964179544769989, 4.06095534746289918622858142520, 4.73688837916018369756851686823, 4.85053597454650241862841687672, 5.66778610808101394820397488795, 5.93421836585419308708142466618, 6.17766391193468089129121537353, 6.35870535421854895138558621178, 6.93137968145181140621046171731, 7.40472637544268798116812868346, 7.87589259388261715397818721514, 7.917225906548906442004130191206, 8.720762709195757263662235507376, 8.857449078695155521324121511884

Graph of the ZZ-function along the critical line