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Properties

Label 4-1280e2-1.1-c1e2-0-18
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 104.465104.465
Root an. cond. 3.197003.19700
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  + 2·9-s + 12·17-s − 25-s − 16·31-s + 12·41-s − 24·47-s − 14·49-s + 8·71-s + 20·73-s + 16·79-s − 5·81-s − 28·89-s + 12·97-s − 16·103-s + 4·113-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.91·17-s − 1/5·25-s − 2.87·31-s + 1.87·41-s − 3.50·47-s − 2·49-s + 0.949·71-s + 2.34·73-s + 1.80·79-s − 5/9·81-s − 2.96·89-s + 1.21·97-s − 1.57·103-s + 0.376·113-s + 1.63·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 104.465104.465
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :1/2,1/2), 1)(4,\ 1638400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3045158432.304515843
L(12)L(\frac12) ≈\approx 2.3045158432.304515843
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
5C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
19C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
61C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
67C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
71C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
89C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{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

−9.816460131092580303730868942707, −9.499169070666755080743966375674, −9.434934158302636330706413414090, −8.428869001446564336675684523140, −8.281400874198584780675458182553, −7.69256064883140028023155388733, −7.62060490790740585039149089420, −7.06159802292879535449280223685, −6.61409216123563902817097371329, −6.10752696829609909612834059768, −5.61183253021110206336438294167, −5.26228028831160479432194375085, −4.93179507836092600893406896408, −4.23081623646308756979369204744, −3.59845659618850537199307707056, −3.41933549884485872791202548018, −2.89836776295036517993281273341, −1.79161102140111418930235135355, −1.63414245439130381187608014171, −0.66125372831476618152163623079, 0.66125372831476618152163623079, 1.63414245439130381187608014171, 1.79161102140111418930235135355, 2.89836776295036517993281273341, 3.41933549884485872791202548018, 3.59845659618850537199307707056, 4.23081623646308756979369204744, 4.93179507836092600893406896408, 5.26228028831160479432194375085, 5.61183253021110206336438294167, 6.10752696829609909612834059768, 6.61409216123563902817097371329, 7.06159802292879535449280223685, 7.62060490790740585039149089420, 7.69256064883140028023155388733, 8.281400874198584780675458182553, 8.428869001446564336675684523140, 9.434934158302636330706413414090, 9.499169070666755080743966375674, 9.816460131092580303730868942707

Graph of the ZZ-function along the critical line