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Properties

Label 4-10e2-1.1-c10e2-0-0
Degree 44
Conductor 100100
Sign 11
Analytic cond. 40.367840.3678
Root an. cond. 2.520622.52062
Motivic weight 1010
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  + 32·2-s − 366·3-s + 512·4-s − 3.75e3·5-s − 1.17e4·6-s − 1.68e4·7-s + 6.69e4·9-s − 1.20e5·10-s − 3.46e5·11-s − 1.87e5·12-s − 4.65e5·13-s − 5.38e5·14-s + 1.37e6·15-s − 2.62e5·16-s + 3.76e6·17-s + 2.14e6·18-s − 1.92e6·20-s + 6.15e6·21-s − 1.10e7·22-s − 1.04e7·23-s + 4.29e6·25-s − 1.48e7·26-s − 2.16e7·27-s − 8.60e6·28-s + 4.39e7·30-s − 2.01e7·31-s − 8.38e6·32-s + ⋯
L(s)  = 1  + 2-s − 1.50·3-s + 1/2·4-s − 6/5·5-s − 1.50·6-s − 1.00·7-s + 1.13·9-s − 6/5·10-s − 2.15·11-s − 0.753·12-s − 1.25·13-s − 1.00·14-s + 1.80·15-s − 1/4·16-s + 2.64·17-s + 1.13·18-s − 3/5·20-s + 1.50·21-s − 2.15·22-s − 1.62·23-s + 0.439·25-s − 1.25·26-s − 1.50·27-s − 0.500·28-s + 1.80·30-s − 0.703·31-s − 1/4·32-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s)2 L(s)=( Λ(11−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+5)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22⋅522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 40.367840.3678
Root analytic conductor: 2.520622.52062
Motivic weight: 1010
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 100, ( :5,5), 1)(4,\ 100,\ (\ :5, 5),\ 1)

Particular Values

L(112)L(\frac{11}{2}) ≈\approx 0.17694157570.1769415757
L(12)L(\frac12) ≈\approx 0.17694157570.1769415757
L(6)L(6) 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)
bad2C2C_2 1−p5T+p9T2 1 - p^{5} T + p^{9} T^{2}
5C2C_2 1+6p4T+p10T2 1 + 6 p^{4} T + p^{10} T^{2}
good3C22C_2^2 1+122pT+7442p2T2+122p11T3+p20T4 1 + 122 p T + 7442 p^{2} T^{2} + 122 p^{11} T^{3} + p^{20} T^{4}
7C22C_2^2 1+2402pT+2884802p2T2+2402p11T3+p20T4 1 + 2402 p T + 2884802 p^{2} T^{2} + 2402 p^{11} T^{3} + p^{20} T^{4}
11C2C_2 (1+173398T+p10T2)2 ( 1 + 173398 T + p^{10} T^{2} )^{2}
13C22C_2^2 1+465246T+108226920258T2+465246p10T3+p20T4 1 + 465246 T + 108226920258 T^{2} + 465246 p^{10} T^{3} + p^{20} T^{4}
17C22C_2^2 1−3760066T+7069048162178T2−3760066p10T3+p20T4 1 - 3760066 T + 7069048162178 T^{2} - 3760066 p^{10} T^{3} + p^{20} T^{4}
19C22C_2^2 1−11048698082002T2+p20T4 1 - 11048698082002 T^{2} + p^{20} T^{4}
23C22C_2^2 1+10456526T+54669467994338T2+10456526p10T3+p20T4 1 + 10456526 T + 54669467994338 T^{2} + 10456526 p^{10} T^{3} + p^{20} T^{4}
29C22C_2^2 1−226828718694802T2+p20T4 1 - 226828718694802 T^{2} + p^{20} T^{4}
31C2C_2 (1+10065998T+p10T2)2 ( 1 + 10065998 T + p^{10} T^{2} )^{2}
37C22C_2^2 1−112775826T+6359193464991138T2−112775826p10T3+p20T4 1 - 112775826 T + 6359193464991138 T^{2} - 112775826 p^{10} T^{3} + p^{20} T^{4}
41C2C_2 (1+153003598T+p10T2)2 ( 1 + 153003598 T + p^{10} T^{2} )^{2}
43C22C_2^2 1−118744914T+7050177300433698T2−118744914p10T3+p20T4 1 - 118744914 T + 7050177300433698 T^{2} - 118744914 p^{10} T^{3} + p^{20} T^{4}
47C22C_2^2 1−344678706T+59401705184917218T2−344678706p10T3+p20T4 1 - 344678706 T + 59401705184917218 T^{2} - 344678706 p^{10} T^{3} + p^{20} T^{4}
53C22C_2^2 1−139826p2T+9775655138p4T2−139826p12T3+p20T4 1 - 139826 p^{2} T + 9775655138 p^{4} T^{2} - 139826 p^{12} T^{3} + p^{20} T^{4}
59C22C_2^2 1−155272002554242p2T2+p20T4 1 - 155272002554242 p^{2} T^{2} + p^{20} T^{4}
61C2C_2 (1−906185802T+p10T2)2 ( 1 - 906185802 T + p^{10} T^{2} )^{2}
67C22C_2^2 1+1924147934T+1851172635958234178T2+1924147934p10T3+p20T4 1 + 1924147934 T + 1851172635958234178 T^{2} + 1924147934 p^{10} T^{3} + p^{20} T^{4}
71C2C_2 (1+3120877598T+p10T2)2 ( 1 + 3120877598 T + p^{10} T^{2} )^{2}
73C22C_2^2 1+1272678526T+809855315270766338T2+1272678526p10T3+p20T4 1 + 1272678526 T + 809855315270766338 T^{2} + 1272678526 p^{10} T^{3} + p^{20} T^{4}
79C22C_2^2 1−15063541390202404802T2+p20T4 1 - 15063541390202404802 T^{2} + p^{20} T^{4}
83C22C_2^2 1+10367644206T+53744023191102685218T2+10367644206p10T3+p20T4 1 + 10367644206 T + 53744023191102685218 T^{2} + 10367644206 p^{10} T^{3} + p^{20} T^{4}
89C22C_2^2 1−1969041775268808802T2+p20T4 1 - 1969041775268808802 T^{2} + p^{20} T^{4}
97C22C_2^2 1+1280722494T+820125053318790018T2+1280722494p10T3+p20T4 1 + 1280722494 T + 820125053318790018 T^{2} + 1280722494 p^{10} T^{3} + p^{20} 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

−19.35384534432907741148630021168, −18.23498356647136324701241640361, −17.09951454350802975732890847444, −16.38129556966235166425389175243, −16.06842489748991662543680974712, −15.31579883164481178863418187891, −14.55785196726634088092852042834, −13.29615959638143144285255479439, −12.69033673523140136186095112993, −11.86043780059560216164986306203, −11.83386948381122037982389703625, −10.26104920901986566020196732926, −10.08791032879372778294701243055, −7.82031297319141331675296873376, −7.35077607274388347352801449420, −5.63607573809750764452176700342, −5.51890087363622651618771930643, −4.11258631128076034722736884796, −2.91353861480390921986173232870, −0.21298830104623525771142697366, 0.21298830104623525771142697366, 2.91353861480390921986173232870, 4.11258631128076034722736884796, 5.51890087363622651618771930643, 5.63607573809750764452176700342, 7.35077607274388347352801449420, 7.82031297319141331675296873376, 10.08791032879372778294701243055, 10.26104920901986566020196732926, 11.83386948381122037982389703625, 11.86043780059560216164986306203, 12.69033673523140136186095112993, 13.29615959638143144285255479439, 14.55785196726634088092852042834, 15.31579883164481178863418187891, 16.06842489748991662543680974712, 16.38129556966235166425389175243, 17.09951454350802975732890847444, 18.23498356647136324701241640361, 19.35384534432907741148630021168

Graph of the ZZ-function along the critical line