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Properties

Label 4-245e2-1.1-c1e2-0-11
Degree 44
Conductor 6002560025
Sign 11
Analytic cond. 3.827243.82724
Root an. cond. 1.398691.39869
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  + 3·4-s + 2·5-s + 5·9-s + 5·16-s − 12·19-s + 6·20-s − 25-s − 14·29-s + 4·31-s + 15·36-s + 10·41-s + 10·45-s − 20·59-s + 14·61-s + 3·64-s − 4·71-s − 36·76-s + 4·79-s + 10·80-s + 16·81-s − 18·89-s − 24·95-s − 3·100-s + 18·101-s − 10·109-s − 42·116-s − 22·121-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s + 5/3·9-s + 5/4·16-s − 2.75·19-s + 1.34·20-s − 1/5·25-s − 2.59·29-s + 0.718·31-s + 5/2·36-s + 1.56·41-s + 1.49·45-s − 2.60·59-s + 1.79·61-s + 3/8·64-s − 0.474·71-s − 4.12·76-s + 0.450·79-s + 1.11·80-s + 16/9·81-s − 1.90·89-s − 2.46·95-s − 0.299·100-s + 1.79·101-s − 0.957·109-s − 3.89·116-s − 2·121-s + ⋯

Functional equation

Λ(s)=(60025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(60025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6002560025    =    52⋅745^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 3.827243.82724
Root analytic conductor: 1.398691.39869
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 60025, ( :1/2,1/2), 1)(4,\ 60025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4430153972.443015397
L(12)L(\frac12) ≈\approx 2.4430153972.443015397
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 1−2T+pT2 1 - 2 T + p T^{2}
7 1 1
good2C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
29C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
41C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
43C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
47C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
61C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
67C22C_2^2 1−109T2+p2T4 1 - 109 T^{2} + p^{2} T^{4}
71C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
83C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
89C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
97C22C_2^2 1+62T2+p2T4 1 + 62 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

−12.46951940122934231732586921988, −11.84681627409977531747499048886, −11.13252314025887071708034118484, −10.98756286558056088332578216807, −10.32375677978690018736264978442, −10.25327295647993626477769852673, −9.351854607884083566221417858624, −9.271730083406293209770252837771, −8.315789301112807317702033567314, −7.73218485914141313641570698096, −7.29763084938530033112741735749, −6.80122903032974038134729394574, −6.21699066979898812751951526873, −6.06457335607048116366724981930, −5.22034242659032869267184379667, −4.22970151780495777256668162325, −3.99128029617419054937583513285, −2.75196144789082061150707603900, −1.96757377625627144750002423327, −1.71885448512894583614551319232, 1.71885448512894583614551319232, 1.96757377625627144750002423327, 2.75196144789082061150707603900, 3.99128029617419054937583513285, 4.22970151780495777256668162325, 5.22034242659032869267184379667, 6.06457335607048116366724981930, 6.21699066979898812751951526873, 6.80122903032974038134729394574, 7.29763084938530033112741735749, 7.73218485914141313641570698096, 8.315789301112807317702033567314, 9.271730083406293209770252837771, 9.351854607884083566221417858624, 10.25327295647993626477769852673, 10.32375677978690018736264978442, 10.98756286558056088332578216807, 11.13252314025887071708034118484, 11.84681627409977531747499048886, 12.46951940122934231732586921988

Graph of the ZZ-function along the critical line