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Properties

Label 4-950e2-1.1-c1e2-0-5
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 + 4·11-s + 16-s + 2·19-s + 10·29-s − 16·31-s − 5·36-s − 16·41-s − 4·44-s + 5·49-s − 30·59-s + 4·61-s − 64-s + 4·71-s − 2·76-s + 20·79-s + 16·81-s + 20·99-s + 4·101-s + 30·109-s − 10·116-s − 10·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s + 1.20·11-s + 1/4·16-s + 0.458·19-s + 1.85·29-s − 2.87·31-s − 5/6·36-s − 2.49·41-s − 0.603·44-s + 5/7·49-s − 3.90·59-s + 0.512·61-s − 1/8·64-s + 0.474·71-s − 0.229·76-s + 2.25·79-s + 16/9·81-s + 2.01·99-s + 0.398·101-s + 2.87·109-s − 0.928·116-s − 0.909·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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 2.2202235722.220223572
L(12)L(\frac12) ≈\approx 2.2202235722.220223572
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 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
13C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
23C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
29C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
31C2C_2 (1+8T+pT2)2 ( 1 + 8 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+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C22C_2^2 1−105T2+p2T4 1 - 105 T^{2} + p^{2} T^{4}
59C2C_2 (1+15T+pT2)2 ( 1 + 15 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67C22C_2^2 1−125T2+p2T4 1 - 125 T^{2} + p^{2} T^{4}
71C2C_2 (1−2T+pT2)2 ( 1 - 2 T + 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 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C22C_2^2 1−190T2+p2T4 1 - 190 T^{2} + 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

−10.20686177070777993702787758356, −9.896540214040990582267120060212, −9.290777812594767207445269516254, −8.993274606774424378511424752687, −8.921527482239301520337262919554, −7.919176603635043067459351098544, −7.908104386178803227334811603934, −7.18071432601313327592947201680, −6.92435317740451826562152346022, −6.47540811791126218630180644646, −6.08546600481128978129866741152, −5.22989644326966415173770998739, −5.06327090117280067815267076119, −4.33044112260669478401994054540, −4.16604537503819153879657329221, −3.35882232367253671756552109964, −3.24569016120247373212380306101, −1.79230948075124349973494254521, −1.73532994630650044859878851120, −0.75062600675756926202476192265, 0.75062600675756926202476192265, 1.73532994630650044859878851120, 1.79230948075124349973494254521, 3.24569016120247373212380306101, 3.35882232367253671756552109964, 4.16604537503819153879657329221, 4.33044112260669478401994054540, 5.06327090117280067815267076119, 5.22989644326966415173770998739, 6.08546600481128978129866741152, 6.47540811791126218630180644646, 6.92435317740451826562152346022, 7.18071432601313327592947201680, 7.908104386178803227334811603934, 7.919176603635043067459351098544, 8.921527482239301520337262919554, 8.993274606774424378511424752687, 9.290777812594767207445269516254, 9.896540214040990582267120060212, 10.20686177070777993702787758356

Graph of the ZZ-function along the critical line