↑

Properties

Label 4-105e2-1.1-c7e2-0-0
Degree 44
Conductor 1102511025
Sign 11
Analytic cond. 1075.861075.86
Root an. cond. 5.727165.72716
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12·2-s + 54·3-s − 20·4-s + 250·5-s − 648·6-s + 686·7-s + 672·8-s + 2.18e3·9-s − 3.00e3·10-s − 1.09e4·11-s − 1.08e3·12-s − 2.79e3·13-s − 8.23e3·14-s + 1.35e4·15-s − 2.73e3·16-s − 8.28e3·17-s − 2.62e4·18-s − 8.09e3·19-s − 5.00e3·20-s + 3.70e4·21-s + 1.31e5·22-s − 9.09e4·23-s + 3.62e4·24-s + 4.68e4·25-s + 3.35e4·26-s + 7.87e4·27-s − 1.37e4·28-s + ⋯
L(s)  = 1  − 1.06·2-s + 1.15·3-s − 0.156·4-s + 0.894·5-s − 1.22·6-s + 0.755·7-s + 0.464·8-s + 9-s − 0.948·10-s − 2.48·11-s − 0.180·12-s − 0.352·13-s − 0.801·14-s + 1.03·15-s − 0.166·16-s − 0.408·17-s − 1.06·18-s − 0.270·19-s − 0.139·20-s + 0.872·21-s + 2.63·22-s − 1.55·23-s + 0.535·24-s + 3/5·25-s + 0.374·26-s + 0.769·27-s − 0.118·28-s + ⋯

Functional equation

Λ(s)=(11025s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(11025s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1102511025    =    32⋅52⋅723^{2} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 1075.861075.86
Root analytic conductor: 5.727165.72716
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 11025, ( :7/2,7/2), 1)(4,\ 11025,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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)
bad3C1C_1 (1−p3T)2 ( 1 - p^{3} T )^{2}
5C1C_1 (1−p3T)2 ( 1 - p^{3} T )^{2}
7C1C_1 (1−p3T)2 ( 1 - p^{3} T )^{2}
good2D4D_{4} 1+3p2T+41p2T2+3p9T3+p14T4 1 + 3 p^{2} T + 41 p^{2} T^{2} + 3 p^{9} T^{3} + p^{14} T^{4}
11D4D_{4} 1+10976T+68803686T2+10976p7T3+p14T4 1 + 10976 T + 68803686 T^{2} + 10976 p^{7} T^{3} + p^{14} T^{4}
13D4D_{4} 1+2796T+126859566T2+2796p7T3+p14T4 1 + 2796 T + 126859566 T^{2} + 2796 p^{7} T^{3} + p^{14} T^{4}
17D4D_{4} 1+8284T+47624614pT2+8284p7T3+p14T4 1 + 8284 T + 47624614 p T^{2} + 8284 p^{7} T^{3} + p^{14} T^{4}
19D4D_{4} 1+8096T+1803475414T2+8096p7T3+p14T4 1 + 8096 T + 1803475414 T^{2} + 8096 p^{7} T^{3} + p^{14} T^{4}
23D4D_{4} 1+90976T+7977553070T2+90976p7T3+p14T4 1 + 90976 T + 7977553070 T^{2} + 90976 p^{7} T^{3} + p^{14} T^{4}
29D4D_{4} 1+12532T+31965988526T2+12532p7T3+p14T4 1 + 12532 T + 31965988526 T^{2} + 12532 p^{7} T^{3} + p^{14} T^{4}
31D4D_{4} 1+117960T+9090248910T2+117960p7T3+p14T4 1 + 117960 T + 9090248910 T^{2} + 117960 p^{7} T^{3} + p^{14} T^{4}
37D4D_{4} 1+174212T+106259379454T2+174212p7T3+p14T4 1 + 174212 T + 106259379454 T^{2} + 174212 p^{7} T^{3} + p^{14} T^{4}
41D4D_{4} 1+492700T+358124883062T2+492700p7T3+p14T4 1 + 492700 T + 358124883062 T^{2} + 492700 p^{7} T^{3} + p^{14} T^{4}
43D4D_{4} 1+661176T+502549980726T2+661176p7T3+p14T4 1 + 661176 T + 502549980726 T^{2} + 661176 p^{7} T^{3} + p^{14} T^{4}
47D4D_{4} 1+1675408T+1714888352190T2+1675408p7T3+p14T4 1 + 1675408 T + 1714888352190 T^{2} + 1675408 p^{7} T^{3} + p^{14} T^{4}
53D4D_{4} 1+555436T−793299419970T2+555436p7T3+p14T4 1 + 555436 T - 793299419970 T^{2} + 555436 p^{7} T^{3} + p^{14} T^{4}
59D4D_{4} 1+3121176T+6648375612854T2+3121176p7T3+p14T4 1 + 3121176 T + 6648375612854 T^{2} + 3121176 p^{7} T^{3} + p^{14} T^{4}
61D4D_{4} 1+511588T+1747585779726T2+511588p7T3+p14T4 1 + 511588 T + 1747585779726 T^{2} + 511588 p^{7} T^{3} + p^{14} T^{4}
67D4D_{4} 1+252728T+12123587827590T2+252728p7T3+p14T4 1 + 252728 T + 12123587827590 T^{2} + 252728 p^{7} T^{3} + p^{14} T^{4}
71D4D_{4} 1+1099336T+18156240823806T2+1099336p7T3+p14T4 1 + 1099336 T + 18156240823806 T^{2} + 1099336 p^{7} T^{3} + p^{14} T^{4}
73D4D_{4} 1−5012588T+28363475914198T2−5012588p7T3+p14T4 1 - 5012588 T + 28363475914198 T^{2} - 5012588 p^{7} T^{3} + p^{14} T^{4}
79D4D_{4} 1−83648T+38379183136222T2−83648p7T3+p14T4 1 - 83648 T + 38379183136222 T^{2} - 83648 p^{7} T^{3} + p^{14} T^{4}
83D4D_{4} 1+4404184T+30645544257030T2+4404184p7T3+p14T4 1 + 4404184 T + 30645544257030 T^{2} + 4404184 p^{7} T^{3} + p^{14} T^{4}
89D4D_{4} 1+4381564T+93146164629590T2+4381564p7T3+p14T4 1 + 4381564 T + 93146164629590 T^{2} + 4381564 p^{7} T^{3} + p^{14} T^{4}
97D4D_{4} 1+8539828T+4428577747174T2+8539828p7T3+p14T4 1 + 8539828 T + 4428577747174 T^{2} + 8539828 p^{7} T^{3} + p^{14} T^{4}
show more
show less
   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

−12.23107327955428078505588443207, −11.41012945784341050027251651536, −10.62583505301402662394441528983, −10.34688336473285712867502854354, −9.701101184568594692011608215984, −9.621349777941533985981367704140, −8.595108553937894798103835727892, −8.465802402483926236510824704457, −7.916065177931350363735154918886, −7.56292576532902279117260788537, −6.64404285394586224726158658695, −5.79593439519103874850396460804, −4.86391845645128256096464571519, −4.79404598866282585148191269664, −3.45991199962996539324213027092, −2.70658491673205986314339038043, −2.01088633621629174042378525279, −1.58871829284713238397027166937, 0, 0, 1.58871829284713238397027166937, 2.01088633621629174042378525279, 2.70658491673205986314339038043, 3.45991199962996539324213027092, 4.79404598866282585148191269664, 4.86391845645128256096464571519, 5.79593439519103874850396460804, 6.64404285394586224726158658695, 7.56292576532902279117260788537, 7.916065177931350363735154918886, 8.465802402483926236510824704457, 8.595108553937894798103835727892, 9.621349777941533985981367704140, 9.701101184568594692011608215984, 10.34688336473285712867502854354, 10.62583505301402662394441528983, 11.41012945784341050027251651536, 12.23107327955428078505588443207

Graph of the ZZ-function along the critical line