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Properties

Label 4-810e2-1.1-c3e2-0-6
Degree 44
Conductor 656100656100
Sign 11
Analytic cond. 2284.032284.03
Root an. cond. 6.913146.91314
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 5·5-s + 22·7-s − 8·8-s − 10·10-s − 12·11-s − 38·13-s + 44·14-s − 16·16-s + 210·17-s − 314·19-s − 24·22-s − 117·23-s − 76·26-s + 66·29-s + 25·31-s + 420·34-s − 110·35-s + 628·37-s − 628·38-s + 40·40-s − 504·41-s − 380·43-s − 234·46-s − 252·47-s + 343·49-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s + 1.18·7-s − 0.353·8-s − 0.316·10-s − 0.328·11-s − 0.810·13-s + 0.839·14-s − 1/4·16-s + 2.99·17-s − 3.79·19-s − 0.232·22-s − 1.06·23-s − 0.573·26-s + 0.422·29-s + 0.144·31-s + 2.11·34-s − 0.531·35-s + 2.79·37-s − 2.68·38-s + 0.158·40-s − 1.91·41-s − 1.34·43-s − 0.750·46-s − 0.782·47-s + 49-s − 0.0155·53-s + ⋯

Functional equation

Λ(s)=(656100s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(656100s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 656100656100    =    22⋅38⋅522^{2} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 2284.032284.03
Root analytic conductor: 6.913146.91314
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 656100, ( :3/2,3/2), 1)(4,\ 656100,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.2067914991.206791499
L(12)L(\frac12) ≈\approx 1.2067914991.206791499
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)
bad2C2C_2 1−pT+p2T2 1 - p T + p^{2} T^{2}
3 1 1
5C2C_2 1+pT+p2T2 1 + p T + p^{2} T^{2}
good7C22C_2^2 1−22T+141T2−22p3T3+p6T4 1 - 22 T + 141 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1+12T−1187T2+12p3T3+p6T4 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}
13C22C_2^2 1+38T−753T2+38p3T3+p6T4 1 + 38 T - 753 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4}
17C2C_2 (1−105T+p3T2)2 ( 1 - 105 T + p^{3} T^{2} )^{2}
19C2C_2 (1+157T+p3T2)2 ( 1 + 157 T + p^{3} T^{2} )^{2}
23C22C_2^2 1+117T+1522T2+117p3T3+p6T4 1 + 117 T + 1522 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4}
29C22C_2^2 1−66T−20033T2−66p3T3+p6T4 1 - 66 T - 20033 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4}
31C22C_2^2 1−25T−29166T2−25p3T3+p6T4 1 - 25 T - 29166 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4}
37C2C_2 (1−314T+p3T2)2 ( 1 - 314 T + p^{3} T^{2} )^{2}
41C22C_2^2 1+504T+185095T2+504p3T3+p6T4 1 + 504 T + 185095 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4}
43C22C_2^2 1+380T+64893T2+380p3T3+p6T4 1 + 380 T + 64893 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4}
47C22C_2^2 1+252T−40319T2+252p3T3+p6T4 1 + 252 T - 40319 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4}
53C2C_2 (1+3T+p3T2)2 ( 1 + 3 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+318T−104255T2+318p3T3+p6T4 1 + 318 T - 104255 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1+293T−141132T2+293p3T3+p6T4 1 + 293 T - 141132 T^{2} + 293 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1−322T−197079T2−322p3T3+p6T4 1 - 322 T - 197079 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1−120T+p3T2)2 ( 1 - 120 T + p^{3} T^{2} )^{2}
73C2C_2 (1−44T+p3T2)2 ( 1 - 44 T + p^{3} T^{2} )^{2}
79C22C_2^2 1+917T+347850T2+917p3T3+p6T4 1 + 917 T + 347850 T^{2} + 917 p^{3} T^{3} + p^{6} T^{4}
83C22C_2^2 1−309T−476306T2−309p3T3+p6T4 1 - 309 T - 476306 T^{2} - 309 p^{3} T^{3} + p^{6} T^{4}
89C2C_2 (1+1272T+p3T2)2 ( 1 + 1272 T + p^{3} T^{2} )^{2}
97C22C_2^2 1+1328T+850911T2+1328p3T3+p6T4 1 + 1328 T + 850911 T^{2} + 1328 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

−10.32310079850714570134728317618, −9.640718110465372155674447483974, −9.402801795636124185785972389093, −8.381840690799742040192029065252, −8.293600106156569565133511348856, −7.938781105939453653378450494171, −7.87710416146303603393738228835, −6.82207966331381536559996766244, −6.68837252113636130259731759208, −5.98568793503900486178271469123, −5.55392083307626242918396739615, −5.17848193674032324778745052750, −4.56634914606467185600025818664, −4.17988434281970642206530107805, −3.97282039075007971402284918646, −3.02315604666529819145753193727, −2.66115128845323111753177665246, −1.84761429409608250972532611833, −1.37078257396740041879656305229, −0.25570659414586468780190549490, 0.25570659414586468780190549490, 1.37078257396740041879656305229, 1.84761429409608250972532611833, 2.66115128845323111753177665246, 3.02315604666529819145753193727, 3.97282039075007971402284918646, 4.17988434281970642206530107805, 4.56634914606467185600025818664, 5.17848193674032324778745052750, 5.55392083307626242918396739615, 5.98568793503900486178271469123, 6.68837252113636130259731759208, 6.82207966331381536559996766244, 7.87710416146303603393738228835, 7.938781105939453653378450494171, 8.293600106156569565133511348856, 8.381840690799742040192029065252, 9.402801795636124185785972389093, 9.640718110465372155674447483974, 10.32310079850714570134728317618

Graph of the ZZ-function along the critical line