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Properties

Label 4-1025-1.1-c1e2-0-0
Degree 44
Conductor 10251025
Sign 11
Analytic cond. 0.06535480.0653548
Root an. cond. 0.5056140.505614
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5-s − 4·9-s + 3·11-s + 5·19-s + 2·20-s − 4·25-s + 11·29-s − 6·31-s + 8·36-s − 41-s − 6·44-s + 4·45-s − 6·49-s − 3·55-s + 59-s − 2·61-s + 8·64-s − 3·71-s − 10·76-s − 15·79-s + 7·81-s − 9·89-s − 5·95-s − 12·99-s + 8·100-s + 15·101-s + ⋯
L(s)  = 1  − 4-s − 0.447·5-s − 4/3·9-s + 0.904·11-s + 1.14·19-s + 0.447·20-s − 4/5·25-s + 2.04·29-s − 1.07·31-s + 4/3·36-s − 0.156·41-s − 0.904·44-s + 0.596·45-s − 6/7·49-s − 0.404·55-s + 0.130·59-s − 0.256·61-s + 64-s − 0.356·71-s − 1.14·76-s − 1.68·79-s + 7/9·81-s − 0.953·89-s − 0.512·95-s − 1.20·99-s + 4/5·100-s + 1.49·101-s + ⋯

Functional equation

Λ(s)=(1025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 10251025    =    52⋅415^{2} \cdot 41
Sign: 11
Analytic conductor: 0.06535480.0653548
Root analytic conductor: 0.5056140.505614
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1025, ( :1/2,1/2), 1)(4,\ 1025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.42221415900.4222141590
L(12)L(\frac12) ≈\approx 0.42221415900.4222141590
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+pT2 1 + T + p T^{2}
41C1C_1×\timesC2C_2 (1+T)(1+pT2) ( 1 + T )( 1 + p T^{2} )
good2C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
3C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
7C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1−5T+pT2)(1+2T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C22C_2^2 1−16T2+p2T4 1 - 16 T^{2} + p^{2} T^{4}
17C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
19C2C_2×\timesC2C_2 (1−6T+pT2)(1+T+pT2) ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} )
23C22C_2^2 1−19T2+p2T4 1 - 19 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−9T+pT2)(1−2T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} )
31C2C_2×\timesC2C_2 (1−4T+pT2)(1+10T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C22C_2^2 1−61T2+p2T4 1 - 61 T^{2} + p^{2} T^{4}
43C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
47C22C_2^2 1−40T2+p2T4 1 - 40 T^{2} + p^{2} T^{4}
53C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−4T+pT2)(1+3T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−6T+pT2)(1+8T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C22C_2^2 1+88T2+p2T4 1 + 88 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1−9T+pT2)(1+12T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} )
73C22C_2^2 1+109T2+p2T4 1 + 109 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1+4T+pT2)(1+11T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} )
83C22C_2^2 1−79T2+p2T4 1 - 79 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+T+pT2)(1+8T+pT2) ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} )
97C22C_2^2 1+152T2+p2T4 1 + 152 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

−14.19454045717374451127091353927, −13.78836223859467724349513577428, −13.02132215935876782760497681759, −12.16321002560880394673395714560, −11.64675702466879730087811532096, −11.19128473596752981447430276521, −10.09738201918885273142298867748, −9.423661061919065844869674749372, −8.748731100307867456017587191214, −8.282454962036726379628037425237, −7.29437775775195108905884201252, −6.23530102487327827298160382161, −5.31271891550307177711333795025, −4.34036075847737870492735412284, −3.22692316031770542346454550404, 3.22692316031770542346454550404, 4.34036075847737870492735412284, 5.31271891550307177711333795025, 6.23530102487327827298160382161, 7.29437775775195108905884201252, 8.282454962036726379628037425237, 8.748731100307867456017587191214, 9.423661061919065844869674749372, 10.09738201918885273142298867748, 11.19128473596752981447430276521, 11.64675702466879730087811532096, 12.16321002560880394673395714560, 13.02132215935876782760497681759, 13.78836223859467724349513577428, 14.19454045717374451127091353927

Graph of the ZZ-function along the critical line