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Properties

Label 4-1134e2-1.1-c1e2-0-13
Degree 44
Conductor 12859561285956
Sign 11
Analytic cond. 81.993681.9936
Root an. cond. 3.009153.00915
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-s − 6·5-s − 4·7-s + 8-s + 6·10-s − 6·11-s + 4·13-s + 4·14-s − 16-s + 4·19-s + 6·22-s + 17·25-s − 4·26-s + 9·29-s + 31-s + 24·35-s − 8·37-s − 4·38-s − 6·40-s + 10·43-s − 6·47-s + 9·49-s − 17·50-s − 3·53-s + 36·55-s − 4·56-s − 9·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.68·5-s − 1.51·7-s + 0.353·8-s + 1.89·10-s − 1.80·11-s + 1.10·13-s + 1.06·14-s − 1/4·16-s + 0.917·19-s + 1.27·22-s + 17/5·25-s − 0.784·26-s + 1.67·29-s + 0.179·31-s + 4.05·35-s − 1.31·37-s − 0.648·38-s − 0.948·40-s + 1.52·43-s − 0.875·47-s + 9/7·49-s − 2.40·50-s − 0.412·53-s + 4.85·55-s − 0.534·56-s − 1.18·58-s + ⋯

Functional equation

Λ(s)=(1285956s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1285956s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12859561285956    =    22⋅38⋅722^{2} \cdot 3^{8} \cdot 7^{2}
Sign: 11
Analytic conductor: 81.993681.9936
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1285956, ( :1/2,1/2), 1)(4,\ 1285956,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.34841943280.3484194328
L(12)L(\frac12) ≈\approx 0.34841943280.3484194328
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+T+T2 1 + T + T^{2}
3 1 1
7C2C_2 1+4T+pT2 1 + 4 T + p T^{2}
good5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C22C_2^2 1−4T+3T2−4pT3+p2T4 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
19C22C_2^2 1−4T−3T2−4pT3+p2T4 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1−9T+52T2−9pT3+p2T4 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−T−30T2−pT3+p2T4 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4}
37C22C_2^2 1+8T+27T2+8pT3+p2T4 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4}
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C22C_2^2 1−10T+57T2−10pT3+p2T4 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+6T−11T2+6pT3+p2T4 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+3T−44T2+3pT3+p2T4 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−3T−50T2−3pT3+p2T4 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−10T+39T2−10pT3+p2T4 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−10T+33T2−10pT3+p2T4 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C22C_2^2 1+2T−69T2+2pT3+p2T4 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1−T−78T2−pT3+p2T4 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4}
83C22C_2^2 1+9T−2T2+9pT3+p2T4 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−6T−53T2−6pT3+p2T4 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−T−96T2−pT3+p2T4 1 - T - 96 T^{2} - p T^{3} + 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

−10.11037854145630395391381806303, −9.653612908057771892577354172587, −9.015294700930763744891030780614, −8.744369563981517876634704711525, −8.205699823767514146455640557572, −8.075290885194319396111210317419, −7.68951186471539957170931728713, −7.28234333246632456654464533604, −6.90395084743673578452748678067, −6.43419274011781789983994184229, −5.89901429984915536176116827754, −5.14221082664712962715647889333, −4.91619895746372605971117492610, −4.13137034474540840853394773361, −3.81420342988359060376885150845, −3.28750892510202225909780605845, −3.07575075472159134007379977860, −2.31352085574566267005931063868, −0.867481232918765214860434098757, −0.41478198035576836475555357687, 0.41478198035576836475555357687, 0.867481232918765214860434098757, 2.31352085574566267005931063868, 3.07575075472159134007379977860, 3.28750892510202225909780605845, 3.81420342988359060376885150845, 4.13137034474540840853394773361, 4.91619895746372605971117492610, 5.14221082664712962715647889333, 5.89901429984915536176116827754, 6.43419274011781789983994184229, 6.90395084743673578452748678067, 7.28234333246632456654464533604, 7.68951186471539957170931728713, 8.075290885194319396111210317419, 8.205699823767514146455640557572, 8.744369563981517876634704711525, 9.015294700930763744891030780614, 9.653612908057771892577354172587, 10.11037854145630395391381806303

Graph of the ZZ-function along the critical line