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Properties

Label 4-3168e2-1.1-c1e2-0-15
Degree 44
Conductor 1003622410036224
Sign 11
Analytic cond. 639.918639.918
Root an. cond. 5.029575.02957
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 2·11-s + 4·13-s − 6·17-s − 6·19-s − 3·23-s + 25-s − 6·29-s + 15·31-s + 37-s − 6·43-s − 8·47-s − 14·49-s − 6·55-s − 13·59-s − 6·61-s − 12·65-s − 3·67-s + 19·71-s − 12·73-s + 18·79-s + 2·83-s + 18·85-s + 3·89-s + 18·95-s − 97-s − 18·101-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.603·11-s + 1.10·13-s − 1.45·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s + 2.69·31-s + 0.164·37-s − 0.914·43-s − 1.16·47-s − 2·49-s − 0.809·55-s − 1.69·59-s − 0.768·61-s − 1.48·65-s − 0.366·67-s + 2.25·71-s − 1.40·73-s + 2.02·79-s + 0.219·83-s + 1.95·85-s + 0.317·89-s + 1.84·95-s − 0.101·97-s − 1.79·101-s + ⋯

Functional equation

Λ(s)=(10036224s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10036224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10036224s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10036224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1003622410036224    =    210⋅34⋅1122^{10} \cdot 3^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 639.918639.918
Root analytic conductor: 5.029575.02957
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 10036224, ( :1/2,1/2), 1)(4,\ 10036224,\ (\ :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)
bad2 1 1
3 1 1
11C1C_1 (1−T)2 ( 1 - T )^{2}
good5C22C_2^2 1+3T+8T2+3pT3+p2T4 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4}
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17D4D_{4} 1+6T+26T2+6pT3+p2T4 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+6T+30T2+6pT3+p2T4 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+3T+10T2+3pT3+p2T4 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+6T+50T2+6pT3+p2T4 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−15T+114T2−15pT3+p2T4 1 - 15 T + 114 T^{2} - 15 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−T+36T2−pT3+p2T4 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4}
41C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
43D4D_{4} 1+6T+78T2+6pT3+p2T4 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4}
47C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
53C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
59D4D_{4} 1+13T+122T2+13pT3+p2T4 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+6T−22T2+6pT3+p2T4 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+3T+98T2+3pT3+p2T4 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−19T+194T2−19pT3+p2T4 1 - 19 T + 194 T^{2} - 19 p T^{3} + p^{2} T^{4}
73C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
79D4D_{4} 1−18T+222T2−18pT3+p2T4 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−2T+14T2−2pT3+p2T4 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−3T+176T2−3pT3+p2T4 1 - 3 T + 176 T^{2} - 3 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+T+156T2+pT3+p2T4 1 + T + 156 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

−8.270014063154563479463727586111, −8.245626168003335252540146068248, −7.76345369951209799085435051763, −7.67157290009868726354504921527, −6.69831114291708554928070266074, −6.59283119029711512261613926079, −6.34755789326942501570946477295, −6.19121182794177442485949515601, −5.28596208732442966384738442766, −4.94702257623900185278320827007, −4.37077969356154167704719650484, −4.27533101352841133875950963983, −3.69127476549459434347909838450, −3.59800863294767558936023093756, −2.80128735646112043636618196900, −2.41524788257273881335545028224, −1.62732435612824663355192277690, −1.29319010062930106062844847153, 0, 0, 1.29319010062930106062844847153, 1.62732435612824663355192277690, 2.41524788257273881335545028224, 2.80128735646112043636618196900, 3.59800863294767558936023093756, 3.69127476549459434347909838450, 4.27533101352841133875950963983, 4.37077969356154167704719650484, 4.94702257623900185278320827007, 5.28596208732442966384738442766, 6.19121182794177442485949515601, 6.34755789326942501570946477295, 6.59283119029711512261613926079, 6.69831114291708554928070266074, 7.67157290009868726354504921527, 7.76345369951209799085435051763, 8.245626168003335252540146068248, 8.270014063154563479463727586111

Graph of the ZZ-function along the critical line