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Properties

Label 4-1710e2-1.1-c1e2-0-14
Degree 44
Conductor 29241002924100
Sign 11
Analytic cond. 186.443186.443
Root an. cond. 3.695183.69518
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  + 2·2-s + 3·4-s + 2·5-s + 4·7-s + 4·8-s + 4·10-s + 4·13-s + 8·14-s + 5·16-s + 2·19-s + 6·20-s + 3·25-s + 8·26-s + 12·28-s + 4·31-s + 6·32-s + 8·35-s + 4·37-s + 4·38-s + 8·40-s + 4·43-s − 2·49-s + 6·50-s + 12·52-s − 12·53-s + 16·56-s + 4·61-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.51·7-s + 1.41·8-s + 1.26·10-s + 1.10·13-s + 2.13·14-s + 5/4·16-s + 0.458·19-s + 1.34·20-s + 3/5·25-s + 1.56·26-s + 2.26·28-s + 0.718·31-s + 1.06·32-s + 1.35·35-s + 0.657·37-s + 0.648·38-s + 1.26·40-s + 0.609·43-s − 2/7·49-s + 0.848·50-s + 1.66·52-s − 1.64·53-s + 2.13·56-s + 0.512·61-s + ⋯

Functional equation

Λ(s)=(2924100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2924100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 29241002924100    =    22⋅34⋅52⋅1922^{2} \cdot 3^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 186.443186.443
Root analytic conductor: 3.695183.69518
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2924100, ( :1/2,1/2), 1)(4,\ 2924100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 11.1429462911.14294629
L(12)L(\frac12) ≈\approx 11.1429462911.14294629
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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
31D4D_{4} 1−4T+54T2−4pT3+p2T4 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−4T+66T2−4pT3+p2T4 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43D4D_{4} 1−4T+78T2−4pT3+p2T4 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
53D4D_{4} 1+12T+94T2+12pT3+p2T4 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+106T2+p2T4 1 + 106 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
67D4D_{4} 1−16T+150T2−16pT3+p2T4 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
73D4D_{4} 1−4T+102T2−4pT3+p2T4 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−4T+150T2−4pT3+p2T4 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+12T+190T2+12pT3+p2T4 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4}
89C22C_2^2 1+130T2+p2T4 1 + 130 T^{2} + p^{2} T^{4}
97D4D_{4} 1−16T+150T2−16pT3+p2T4 1 - 16 T + 150 T^{2} - 16 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

−9.572102116122105973925851014545, −9.215396895388497294359717706153, −8.505486021083172175158181775082, −8.358481985376727830261517178709, −7.75395437650156839668117403755, −7.71851644262336002436865439342, −6.78250190868645198460136723582, −6.75375601368792312579748989962, −6.02217355508734028869513452271, −5.98342385044188577388055379982, −5.22677938315602569812568055562, −5.17957824185817504360742572966, −4.51618609295393231917184584628, −4.37743777981621570503957637080, −3.52087112438395697347711270639, −3.37230508288646908882310828362, −2.41080308782883156323643119888, −2.29752280015848155476774902238, −1.31873407300452199612511982994, −1.26220932573151456838490135346, 1.26220932573151456838490135346, 1.31873407300452199612511982994, 2.29752280015848155476774902238, 2.41080308782883156323643119888, 3.37230508288646908882310828362, 3.52087112438395697347711270639, 4.37743777981621570503957637080, 4.51618609295393231917184584628, 5.17957824185817504360742572966, 5.22677938315602569812568055562, 5.98342385044188577388055379982, 6.02217355508734028869513452271, 6.75375601368792312579748989962, 6.78250190868645198460136723582, 7.71851644262336002436865439342, 7.75395437650156839668117403755, 8.358481985376727830261517178709, 8.505486021083172175158181775082, 9.215396895388497294359717706153, 9.572102116122105973925851014545

Graph of the ZZ-function along the critical line