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Properties

Label 4-648675-1.1-c1e2-0-6
Degree 44
Conductor 648675648675
Sign 11
Analytic cond. 41.360041.3600
Root an. cond. 2.535972.53597
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·4-s − 4·7-s + 9-s + 3·12-s + 5·16-s − 16·19-s + 4·21-s + 25-s − 27-s + 12·28-s + 2·31-s − 3·36-s + 16·37-s − 5·48-s − 2·49-s + 16·57-s − 28·61-s − 4·63-s − 3·64-s + 4·67-s − 32·73-s − 75-s + 48·76-s + 81-s − 12·84-s − 2·93-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s − 1.51·7-s + 1/3·9-s + 0.866·12-s + 5/4·16-s − 3.67·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 2.26·28-s + 0.359·31-s − 1/2·36-s + 2.63·37-s − 0.721·48-s − 2/7·49-s + 2.11·57-s − 3.58·61-s − 0.503·63-s − 3/8·64-s + 0.488·67-s − 3.74·73-s − 0.115·75-s + 5.50·76-s + 1/9·81-s − 1.30·84-s − 0.207·93-s + ⋯

Functional equation

Λ(s)=(648675s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(648675s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 648675648675    =    33⋅52⋅3123^{3} \cdot 5^{2} \cdot 31^{2}
Sign: 11
Analytic conductor: 41.360041.3600
Root analytic conductor: 2.535972.53597
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 648675, ( :1/2,1/2), 1)(4,\ 648675,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3C1C_1 1+T 1 + T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
31C1C_1 (1−T)2 ( 1 - T )^{2}
good2C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
7C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
61C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
67C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
73C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
89C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
97C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
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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

−7.84532589169742215840335224293, −7.68128969255594037111738820978, −6.77225946146642489618389989901, −6.36573875529349558957576705485, −6.13319511217733986766315838164, −5.87102863181679820049643029680, −4.85221518213525471918852830380, −4.62476804594428139253779251777, −4.15549500339935073670126654606, −3.85399261511576687175772764034, −2.96653770006789309828242375320, −2.48347553325427240017401962963, −1.38137651049014592943287717063, 0, 0, 1.38137651049014592943287717063, 2.48347553325427240017401962963, 2.96653770006789309828242375320, 3.85399261511576687175772764034, 4.15549500339935073670126654606, 4.62476804594428139253779251777, 4.85221518213525471918852830380, 5.87102863181679820049643029680, 6.13319511217733986766315838164, 6.36573875529349558957576705485, 6.77225946146642489618389989901, 7.68128969255594037111738820978, 7.84532589169742215840335224293

Graph of the ZZ-function along the critical line