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Properties

Label 4-1710e2-1.1-c3e2-0-8
Degree 44
Conductor 29241002924100
Sign 11
Analytic cond. 10179.410179.4
Root an. cond. 10.044510.0445
Motivic weight 33
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  + 4·2-s + 12·4-s + 10·5-s − 24·7-s + 32·8-s + 40·10-s − 24·11-s − 76·13-s − 96·14-s + 80·16-s + 84·17-s − 38·19-s + 120·20-s − 96·22-s + 136·23-s + 75·25-s − 304·26-s − 288·28-s + 20·29-s − 128·31-s + 192·32-s + 336·34-s − 240·35-s − 188·37-s − 152·38-s + 320·40-s − 188·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.29·7-s + 1.41·8-s + 1.26·10-s − 0.657·11-s − 1.62·13-s − 1.83·14-s + 5/4·16-s + 1.19·17-s − 0.458·19-s + 1.34·20-s − 0.930·22-s + 1.23·23-s + 3/5·25-s − 2.29·26-s − 1.94·28-s + 0.128·29-s − 0.741·31-s + 1.06·32-s + 1.69·34-s − 1.15·35-s − 0.835·37-s − 0.648·38-s + 1.26·40-s − 0.716·41-s + ⋯

Functional equation

Λ(s)=(2924100s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2924100s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{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: 10179.410179.4
Root analytic conductor: 10.044510.0445
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2924100, ( :3/2,3/2), 1)(4,\ 2924100,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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−pT)2 ( 1 - p T )^{2}
3 1 1
5C1C_1 (1−pT)2 ( 1 - p T )^{2}
19C1C_1 (1+pT)2 ( 1 + p T )^{2}
good7D4D_{4} 1+24T+598T2+24p3T3+p6T4 1 + 24 T + 598 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+24T+718T2+24p3T3+p6T4 1 + 24 T + 718 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+76T+5606T2+76p3T3+p6T4 1 + 76 T + 5606 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−84T+10662T2−84p3T3+p6T4 1 - 84 T + 10662 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−136T+25246T2−136p3T3+p6T4 1 - 136 T + 25246 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−20T+46790T2−20p3T3+p6T4 1 - 20 T + 46790 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+128T+59966T2+128p3T3+p6T4 1 + 128 T + 59966 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+188T+1566pT2+188p3T3+p6T4 1 + 188 T + 1566 p T^{2} + 188 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+188T+144590T2+188p3T3+p6T4 1 + 188 T + 144590 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+328T+118862T2+328p3T3+p6T4 1 + 328 T + 118862 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+120T+177838T2+120p3T3+p6T4 1 + 120 T + 177838 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+300T+19582T2+300p3T3+p6T4 1 + 300 T + 19582 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+656T+217670T2+656p3T3+p6T4 1 + 656 T + 217670 T^{2} + 656 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+436T+408686T2+436p3T3+p6T4 1 + 436 T + 408686 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+728T+688550T2+728p3T3+p6T4 1 + 728 T + 688550 T^{2} + 728 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+624T+679534T2+624p3T3+p6T4 1 + 624 T + 679534 T^{2} + 624 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+284T+797270T2+284p3T3+p6T4 1 + 284 T + 797270 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−128T−280258T2−128p3T3+p6T4 1 - 128 T - 280258 T^{2} - 128 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1712T+1816918T2+1712p3T3+p6T4 1 + 1712 T + 1816918 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+1356T+1841550T2+1356p3T3+p6T4 1 + 1356 T + 1841550 T^{2} + 1356 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−788T+1551614T2−788p3T3+p6T4 1 - 788 T + 1551614 T^{2} - 788 p^{3} T^{3} + p^{6} 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

−8.741056859997855473423931196681, −8.405893146272842788303624674398, −7.71718961459527953971850953805, −7.40785290704074470425158174299, −6.89618750774710624663477750526, −6.88273685644190864594313057689, −6.07668786870420372063425359208, −6.01718236853581306114667825625, −5.42806126622887706801604154001, −5.08291140507087395904824827905, −4.76593292222870250329319112974, −4.34513324285721521640920912394, −3.40979821303556915461579134194, −3.31532923697428172217110536540, −2.72179871476417919643837523251, −2.61733473599412061525438083058, −1.57817603426194525628082575579, −1.48229082892597641386653894239, 0, 0, 1.48229082892597641386653894239, 1.57817603426194525628082575579, 2.61733473599412061525438083058, 2.72179871476417919643837523251, 3.31532923697428172217110536540, 3.40979821303556915461579134194, 4.34513324285721521640920912394, 4.76593292222870250329319112974, 5.08291140507087395904824827905, 5.42806126622887706801604154001, 6.01718236853581306114667825625, 6.07668786870420372063425359208, 6.88273685644190864594313057689, 6.89618750774710624663477750526, 7.40785290704074470425158174299, 7.71718961459527953971850953805, 8.405893146272842788303624674398, 8.741056859997855473423931196681

Graph of the ZZ-function along the critical line