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Properties

Label 4-5733e2-1.1-c1e2-0-7
Degree 44
Conductor 3286728932867289
Sign 11
Analytic cond. 2095.642095.64
Root an. cond. 6.765966.76596
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4·4-s + 3·8-s + 6·11-s − 2·13-s + 3·16-s + 6·17-s + 6·19-s + 18·22-s + 12·23-s − 5·25-s − 6·26-s + 10·31-s + 6·32-s + 18·34-s − 4·37-s + 18·38-s − 16·43-s + 24·44-s + 36·46-s + 6·47-s − 15·50-s − 8·52-s + 6·53-s − 6·59-s + 6·61-s + 30·62-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s + 1.06·8-s + 1.80·11-s − 0.554·13-s + 3/4·16-s + 1.45·17-s + 1.37·19-s + 3.83·22-s + 2.50·23-s − 25-s − 1.17·26-s + 1.79·31-s + 1.06·32-s + 3.08·34-s − 0.657·37-s + 2.91·38-s − 2.43·43-s + 3.61·44-s + 5.30·46-s + 0.875·47-s − 2.12·50-s − 1.10·52-s + 0.824·53-s − 0.781·59-s + 0.768·61-s + 3.81·62-s + ⋯

Functional equation

Λ(s)=(32867289s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 32867289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(32867289s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 32867289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3286728932867289    =    34⋅74⋅1323^{4} \cdot 7^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 2095.642095.64
Root analytic conductor: 6.765966.76596
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 32867289, ( :1/2,1/2), 1)(4,\ 32867289,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 15.8481544715.84815447
L(12)L(\frac12) ≈\approx 15.8481544715.84815447
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)
bad3 1 1
7 1 1
13C1C_1 (1+T)2 ( 1 + T )^{2}
good2C22C_2^2 1−3T+5T2−3pT3+p2T4 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4}
5C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
17D4D_{4} 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
19C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
23D4D_{4} 1−12T+77T2−12pT3+p2T4 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
31C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
37D4D_{4} 1+4T+33T2+4pT3+p2T4 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
43C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
47D4D_{4} 1−6T+83T2−6pT3+p2T4 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−6T+95T2−6pT3+p2T4 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+6T+107T2+6pT3+p2T4 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4}
61C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
67C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
71C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
73D4D_{4} 1−8T+117T2−8pT3+p2T4 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−8T+129T2−8pT3+p2T4 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1+173T2+p2T4 1 + 173 T^{2} + p^{2} T^{4}
97D4D_{4} 1+8T+30T2+8pT3+p2T4 1 + 8 T + 30 T^{2} + 8 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

−8.270612109086063232287708959359, −7.69629757132774707611651376452, −7.39496068555586426443115287178, −7.17955611398549923853908406109, −6.62305805131218474681205173489, −6.52839518284144799122193282594, −5.86322008664896671218046157369, −5.81398570354841287006827766299, −5.17501842036711965255703147720, −5.04922240682503243801843755772, −4.69777243122186607431825797567, −4.40802129489538029928248668667, −3.79572697239028377233237807751, −3.50998178364644846807811630612, −3.16810263603139382652102846667, −3.11108144127331549652676534880, −2.31518502414556948623382678579, −1.70417661669501108230058458565, −1.05093587368298752338815604664, −0.883430952743246814743964384621, 0.883430952743246814743964384621, 1.05093587368298752338815604664, 1.70417661669501108230058458565, 2.31518502414556948623382678579, 3.11108144127331549652676534880, 3.16810263603139382652102846667, 3.50998178364644846807811630612, 3.79572697239028377233237807751, 4.40802129489538029928248668667, 4.69777243122186607431825797567, 5.04922240682503243801843755772, 5.17501842036711965255703147720, 5.81398570354841287006827766299, 5.86322008664896671218046157369, 6.52839518284144799122193282594, 6.62305805131218474681205173489, 7.17955611398549923853908406109, 7.39496068555586426443115287178, 7.69629757132774707611651376452, 8.270612109086063232287708959359

Graph of the ZZ-function along the critical line