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Properties

Label 4-312e2-1.1-c1e2-0-11
Degree 44
Conductor 9734497344
Sign 11
Analytic cond. 6.206736.20673
Root an. cond. 1.578391.57839
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  + 9-s + 6·13-s + 4·17-s − 6·25-s + 4·29-s + 2·49-s + 4·53-s − 4·61-s − 24·79-s + 81-s + 4·101-s + 8·103-s + 16·107-s + 20·113-s + 6·117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 23·169-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.66·13-s + 0.970·17-s − 6/5·25-s + 0.742·29-s + 2/7·49-s + 0.549·53-s − 0.512·61-s − 2.70·79-s + 1/9·81-s + 0.398·101-s + 0.788·103-s + 1.54·107-s + 1.88·113-s + 0.554·117-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 6.206736.20673
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 97344, ( :1/2,1/2), 1)(4,\ 97344,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7191153261.719115326
L(12)L(\frac12) ≈\approx 1.7191153261.719115326
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)
bad2 1 1
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
13C2C_2 1−6T+pT2 1 - 6 T + p T^{2}
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
31C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
59C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
71C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+8T+pT2)(1+16T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} )
83C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
89C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
97C22C_2^2 1+126T2+p2T4 1 + 126 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

−9.699438769496114402033257178004, −8.993860452341910821794806769078, −8.589788743802462385769043811054, −8.215799634359465414661645965473, −7.53440663921409189155491314773, −7.22581679137061663285206796479, −6.43986804523822436911164629458, −5.94124585537876212909233098247, −5.67153147242154355347872008580, −4.80024266791599446847212426264, −4.18301772950860384666382204939, −3.59866243929372867606894540722, −3.03644367581316318535608327538, −1.94677222991202967287945155352, −1.07453461325392778034782218303, 1.07453461325392778034782218303, 1.94677222991202967287945155352, 3.03644367581316318535608327538, 3.59866243929372867606894540722, 4.18301772950860384666382204939, 4.80024266791599446847212426264, 5.67153147242154355347872008580, 5.94124585537876212909233098247, 6.43986804523822436911164629458, 7.22581679137061663285206796479, 7.53440663921409189155491314773, 8.215799634359465414661645965473, 8.589788743802462385769043811054, 8.993860452341910821794806769078, 9.699438769496114402033257178004

Graph of the ZZ-function along the critical line