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Properties

Label 4-132300-1.1-c1e2-0-0
Degree 44
Conductor 132300132300
Sign 11
Analytic cond. 8.435568.43556
Root an. cond. 1.704231.70423
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  − 3-s − 4-s − 5-s − 5·7-s + 9-s + 12-s + 15-s + 16-s + 20-s + 5·21-s − 4·25-s − 27-s + 5·28-s + 5·35-s − 36-s − 5·37-s − 10·41-s − 15·43-s − 45-s + 47-s − 48-s + 18·49-s + 20·59-s − 60-s − 5·63-s − 64-s + 4·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.223·20-s + 1.09·21-s − 4/5·25-s − 0.192·27-s + 0.944·28-s + 0.845·35-s − 1/6·36-s − 0.821·37-s − 1.56·41-s − 2.28·43-s − 0.149·45-s + 0.145·47-s − 0.144·48-s + 18/7·49-s + 2.60·59-s − 0.129·60-s − 0.629·63-s − 1/8·64-s + 0.461·75-s + ⋯

Functional equation

Λ(s)=(132300s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(132300s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 132300132300    =    22⋅33⋅52⋅722^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 8.435568.43556
Root analytic conductor: 1.704231.70423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 132300, ( :1/2,1/2), 1)(4,\ 132300,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.43790277940.4379027794
L(12)L(\frac12) ≈\approx 0.43790277940.4379027794
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}
3C1C_1 1+T 1 + T
5C2C_2 1+T+pT2 1 + T + p T^{2}
7C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good11C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
13C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
17C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
29C22C_2^2 1−16T2+p2T4 1 - 16 T^{2} + p^{2} T^{4}
31C22C_2^2 1+32T2+p2T4 1 + 32 T^{2} + p^{2} T^{4}
37C2C_2×\timesC2C_2 (1+T+pT2)(1+4T+pT2) ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} )
41C2C_2×\timesC2C_2 (1+pT2)(1+10T+pT2) ( 1 + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2×\timesC2C_2 (1+4T+pT2)(1+11T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−8T+pT2)(1+7T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} )
53C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−14T+pT2)(1−6T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} )
61C22C_2^2 1+79T2+p2T4 1 + 79 T^{2} + p^{2} T^{4}
67C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−3T+pT2)(1+12T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} )
83C2C_2×\timesC2C_2 (1−12T+pT2)(1−7T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 7 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−12T+pT2)(1+7T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 7 T + p T^{2} )
97C22C_2^2 1−50T2+p2T4 1 - 50 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

−9.607208471564681200929939698031, −8.848816285099318986087744849626, −8.505698999019485589977521256618, −7.984567411096763444580986362622, −7.13682700831533312015440131557, −6.94316955660584393926016610480, −6.41578453280906087258831566326, −5.88874387429572538588576714330, −5.30963037576648192115197073588, −4.81816101464795294409726127939, −3.91421469398545160810260207237, −3.60792296384436066119156884197, −3.03950771406390117947679780329, −1.93991427194705757404501644184, −0.46134343672254373040015387019, 0.46134343672254373040015387019, 1.93991427194705757404501644184, 3.03950771406390117947679780329, 3.60792296384436066119156884197, 3.91421469398545160810260207237, 4.81816101464795294409726127939, 5.30963037576648192115197073588, 5.88874387429572538588576714330, 6.41578453280906087258831566326, 6.94316955660584393926016610480, 7.13682700831533312015440131557, 7.984567411096763444580986362622, 8.505698999019485589977521256618, 8.848816285099318986087744849626, 9.607208471564681200929939698031

Graph of the ZZ-function along the critical line