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Properties

Label 4-95e2-1.1-c1e2-0-3
Degree 44
Conductor 90259025
Sign 11
Analytic cond. 0.5754410.575441
Root an. cond. 0.8709640.870964
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  + 2·3-s + 2·4-s + 5-s − 8·7-s + 3·9-s + 6·11-s + 4·12-s − 2·13-s + 2·15-s − 6·17-s − 7·19-s + 2·20-s − 16·21-s + 10·27-s − 16·28-s + 3·29-s − 14·31-s + 12·33-s − 8·35-s + 6·36-s + 16·37-s − 4·39-s + 6·41-s + 4·43-s + 12·44-s + 3·45-s − 6·47-s + ⋯
L(s)  = 1  + 1.15·3-s + 4-s + 0.447·5-s − 3.02·7-s + 9-s + 1.80·11-s + 1.15·12-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 1.60·19-s + 0.447·20-s − 3.49·21-s + 1.92·27-s − 3.02·28-s + 0.557·29-s − 2.51·31-s + 2.08·33-s − 1.35·35-s + 36-s + 2.63·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 1.80·44-s + 0.447·45-s − 0.875·47-s + ⋯

Functional equation

Λ(s)=(9025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 90259025    =    52⋅1925^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 0.5754410.575441
Root analytic conductor: 0.8709640.870964
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 9025, ( :1/2,1/2), 1)(4,\ 9025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3867737191.386773719
L(12)L(\frac12) ≈\approx 1.3867737191.386773719
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)
bad5C2C_2 1−T+T2 1 - T + T^{2}
19C2C_2 1+7T+pT2 1 + 7 T + p T^{2}
good2C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
3C22C_2^2 1−2T+T2−2pT3+p2T4 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4}
7C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1−5T+pT2)(1+7T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C22C_2^2 1+6T+19T2+6pT3+p2T4 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C22C_2^2 1−3T−20T2−3pT3+p2T4 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4}
31C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C22C_2^2 1−6T−5T2−6pT3+p2T4 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−4T−27T2−4pT3+p2T4 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+6T−11T2+6pT3+p2T4 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−15T+166T2−15pT3+p2T4 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+5T−36T2+5pT3+p2T4 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+2T−63T2+2pT3+p2T4 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−3T−62T2−3pT3+p2T4 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+8T−9T2+8pT3+p2T4 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+5T−54T2+5pT3+p2T4 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4}
83C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
89C22C_2^2 1−15T+136T2−15pT3+p2T4 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+8T−33T2+8pT3+p2T4 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4}
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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

−14.45727815431112247704379615372, −13.46187718623727369698285947222, −13.23941662520608353357695345489, −12.79406976587063842094663671018, −12.40959478732948953346559269490, −11.74685280654655406145570868760, −10.80327490361199919615475316806, −10.55286062112174331753025360208, −9.547593772168487256781364363024, −9.280394832524382960342480921885, −9.239927903917167007568240168427, −8.323530410914496101847811241859, −7.03575539981179466954492587892, −6.93957105375992591277788891737, −6.33318627269780152687854974749, −6.12009696796375254811739836851, −4.25245972936901947248198134348, −3.77337017171378786291803365910, −2.76150367300491270317309970663, −2.29937073169802967093938323200, 2.29937073169802967093938323200, 2.76150367300491270317309970663, 3.77337017171378786291803365910, 4.25245972936901947248198134348, 6.12009696796375254811739836851, 6.33318627269780152687854974749, 6.93957105375992591277788891737, 7.03575539981179466954492587892, 8.323530410914496101847811241859, 9.239927903917167007568240168427, 9.280394832524382960342480921885, 9.547593772168487256781364363024, 10.55286062112174331753025360208, 10.80327490361199919615475316806, 11.74685280654655406145570868760, 12.40959478732948953346559269490, 12.79406976587063842094663671018, 13.23941662520608353357695345489, 13.46187718623727369698285947222, 14.45727815431112247704379615372

Graph of the ZZ-function along the critical line