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Properties

Label 4-507e2-1.1-c1e2-0-11
Degree 44
Conductor 257049257049
Sign 11
Analytic cond. 16.389616.3896
Root an. cond. 2.012062.01206
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  − 2·3-s + 4·4-s + 3·9-s − 8·12-s + 12·16-s − 12·23-s − 2·25-s − 4·27-s + 12·29-s + 12·36-s − 2·43-s − 24·48-s + 11·49-s + 24·53-s + 2·61-s + 32·64-s + 24·69-s + 4·75-s − 22·79-s + 5·81-s − 24·87-s − 48·92-s − 8·100-s − 36·101-s − 2·103-s − 12·107-s − 16·108-s + ⋯
L(s)  = 1  − 1.15·3-s + 2·4-s + 9-s − 2.30·12-s + 3·16-s − 2.50·23-s − 2/5·25-s − 0.769·27-s + 2.22·29-s + 2·36-s − 0.304·43-s − 3.46·48-s + 11/7·49-s + 3.29·53-s + 0.256·61-s + 4·64-s + 2.88·69-s + 0.461·75-s − 2.47·79-s + 5/9·81-s − 2.57·87-s − 5.00·92-s − 4/5·100-s − 3.58·101-s − 0.197·103-s − 1.16·107-s − 1.53·108-s + ⋯

Functional equation

Λ(s)=(257049s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(257049s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 257049257049    =    32⋅1343^{2} \cdot 13^{4}
Sign: 11
Analytic conductor: 16.389616.3896
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 257049, ( :1/2,1/2), 1)(4,\ 257049,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0429594192.042959419
L(12)L(\frac12) ≈\approx 2.0429594192.042959419
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)
bad3C1C_1 (1+T)2 ( 1 + T )^{2}
13 1 1
good2C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
7C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
37C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
41C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
53C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
59C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
61C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
67C22C_2^2 1−59T2+p2T4 1 - 59 T^{2} + p^{2} T^{4}
71C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
73C2C_2 (1−17T+pT2)(1+17T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
83C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
89C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
97C2C_2 (1−19T+pT2)(1+19T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 19 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

−10.99022168541201805180365368170, −10.88459825257715122944784025758, −10.25079254161265134680590827031, −9.983196001486525948829947256632, −9.912383991182912542351654698313, −8.768005058525067394405677337667, −8.229261947696721318114790512853, −7.967477421025086437974286660438, −7.21821716447865331345990610727, −6.95468585242479492883352039003, −6.62915330582822778215013936478, −5.86167570820491382439076694934, −5.83579902619251598800908848119, −5.37784818348028851549801574913, −4.26936580159281048467713232171, −4.05798604057711294781276368355, −3.07939552385925686329295464339, −2.41577920338475777288576920701, −1.86463488811903134360907728775, −0.936485251864389765306096879002, 0.936485251864389765306096879002, 1.86463488811903134360907728775, 2.41577920338475777288576920701, 3.07939552385925686329295464339, 4.05798604057711294781276368355, 4.26936580159281048467713232171, 5.37784818348028851549801574913, 5.83579902619251598800908848119, 5.86167570820491382439076694934, 6.62915330582822778215013936478, 6.95468585242479492883352039003, 7.21821716447865331345990610727, 7.967477421025086437974286660438, 8.229261947696721318114790512853, 8.768005058525067394405677337667, 9.912383991182912542351654698313, 9.983196001486525948829947256632, 10.25079254161265134680590827031, 10.88459825257715122944784025758, 10.99022168541201805180365368170

Graph of the ZZ-function along the critical line