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Properties

Label 4-950e2-1.1-c1e2-0-10
Degree 44
Conductor 902500902500
Sign 11
Analytic cond. 57.544157.5441
Root an. cond. 2.754232.75423
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-s + 5·9-s + 16-s + 2·19-s + 10·29-s + 20·31-s − 5·36-s + 4·41-s + 13·49-s + 14·59-s − 8·61-s − 64-s − 2·76-s + 20·79-s + 16·81-s + 20·89-s − 8·101-s − 26·109-s − 10·116-s − 22·121-s − 20·124-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s + 1/4·16-s + 0.458·19-s + 1.85·29-s + 3.59·31-s − 5/6·36-s + 0.624·41-s + 13/7·49-s + 1.82·59-s − 1.02·61-s − 1/8·64-s − 0.229·76-s + 2.25·79-s + 16/9·81-s + 2.11·89-s − 0.796·101-s − 2.49·109-s − 0.928·116-s − 2·121-s − 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(902500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(902500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 902500902500    =    22⋅54⋅1922^{2} \cdot 5^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 57.544157.5441
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 902500, ( :1/2,1/2), 1)(4,\ 902500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6079588092.607958809
L(12)L(\frac12) ≈\approx 2.6079588092.607958809
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}
5 1 1
19C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C22C_2^2 1−17T2+p2T4 1 - 17 T^{2} + p^{2} T^{4}
17C22C_2^2 1+15T2+p2T4 1 + 15 T^{2} + p^{2} T^{4}
23C22C_2^2 1−21T2+p2T4 1 - 21 T^{2} + p^{2} T^{4}
29C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
31C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
43C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
59C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
61C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
67C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1−65T2+p2T4 1 - 65 T^{2} + p^{2} T^{4}
79C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
83C22C_2^2 1−162T2+p2T4 1 - 162 T^{2} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
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

−10.18045213609076050203781860461, −9.887229082940113276627946832169, −9.521732143685579722395560465754, −9.040084914755131211440845574866, −8.570767921482245233489888072917, −8.146375880621783584188629201985, −7.73536537914502029469829826164, −7.42322210336377208774835263203, −6.69605789025234536577527003526, −6.48597861289769798939981143583, −6.16558918619354528010285186770, −5.23414264969154629438542205541, −4.93342210299783507043727648191, −4.53755516569122730922776405965, −4.00081605933044469909938997732, −3.68868125335052577976451741625, −2.58516429259994352815771479682, −2.54915768872178716372889155151, −1.07694327410406104773496674464, −1.06935986225685257853468118180, 1.06935986225685257853468118180, 1.07694327410406104773496674464, 2.54915768872178716372889155151, 2.58516429259994352815771479682, 3.68868125335052577976451741625, 4.00081605933044469909938997732, 4.53755516569122730922776405965, 4.93342210299783507043727648191, 5.23414264969154629438542205541, 6.16558918619354528010285186770, 6.48597861289769798939981143583, 6.69605789025234536577527003526, 7.42322210336377208774835263203, 7.73536537914502029469829826164, 8.146375880621783584188629201985, 8.570767921482245233489888072917, 9.040084914755131211440845574866, 9.521732143685579722395560465754, 9.887229082940113276627946832169, 10.18045213609076050203781860461

Graph of the ZZ-function along the critical line