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Properties

Label 4-2352e2-1.1-c3e2-0-3
Degree 44
Conductor 55319045531904
Sign 11
Analytic cond. 19257.819257.8
Root an. cond. 11.780111.7801
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  + 6·3-s + 18·5-s + 27·9-s − 58·11-s + 48·13-s + 108·15-s − 6·17-s − 84·19-s − 6·23-s + 130·25-s + 108·27-s − 104·29-s − 252·31-s − 348·33-s + 224·37-s + 288·39-s + 438·41-s + 776·43-s + 486·45-s + 240·47-s − 36·51-s + 1.08e3·53-s − 1.04e3·55-s − 504·57-s − 312·59-s − 660·61-s + 864·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.60·5-s + 9-s − 1.58·11-s + 1.02·13-s + 1.85·15-s − 0.0856·17-s − 1.01·19-s − 0.0543·23-s + 1.03·25-s + 0.769·27-s − 0.665·29-s − 1.46·31-s − 1.83·33-s + 0.995·37-s + 1.18·39-s + 1.66·41-s + 2.75·43-s + 1.60·45-s + 0.744·47-s − 0.0988·51-s + 2.79·53-s − 2.55·55-s − 1.17·57-s − 0.688·59-s − 1.38·61-s + 1.64·65-s + ⋯

Functional equation

Λ(s)=(5531904s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(5531904s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 55319045531904    =    28⋅32⋅742^{8} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 19257.819257.8
Root analytic conductor: 11.780111.7801
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 5531904, ( :3/2,3/2), 1)(4,\ 5531904,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 9.5288669009.528866900
L(12)L(\frac12) ≈\approx 9.5288669009.528866900
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)
bad2 1 1
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
7 1 1
good5D4D_{4} 1−18T+194T2−18p3T3+p6T4 1 - 18 T + 194 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+58T+2270T2+58p3T3+p6T4 1 + 58 T + 2270 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−48T+2778T2−48p3T3+p6T4 1 - 48 T + 2778 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+6T+9698T2+6p3T3+p6T4 1 + 6 T + 9698 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+84T+1782T2+84p3T3+p6T4 1 + 84 T + 1782 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+6T−6482T2+6p3T3+p6T4 1 + 6 T - 6482 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+104T+46550T2+104p3T3+p6T4 1 + 104 T + 46550 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+252T+74910T2+252p3T3+p6T4 1 + 252 T + 74910 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−224T+108918T2−224p3T3+p6T4 1 - 224 T + 108918 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−438T+113330T2−438p3T3+p6T4 1 - 438 T + 113330 T^{2} - 438 p^{3} T^{3} + p^{6} T^{4}
43C2C_2 (1−388T+p3T2)2 ( 1 - 388 T + p^{3} T^{2} )^{2}
47D4D_{4} 1−240T+81758T2−240p3T3+p6T4 1 - 240 T + 81758 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−1080T+584422T2−1080p3T3+p6T4 1 - 1080 T + 584422 T^{2} - 1080 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+312T+400022T2+312p3T3+p6T4 1 + 312 T + 400022 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+660T+496554T2+660p3T3+p6T4 1 + 660 T + 496554 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−396T+399062T2−396p3T3+p6T4 1 - 396 T + 399062 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+66T+360574T2+66p3T3+p6T4 1 + 66 T + 360574 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−972T+969842T2−972p3T3+p6T4 1 - 972 T + 969842 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−860T+1126590T2−860p3T3+p6T4 1 - 860 T + 1126590 T^{2} - 860 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+2520T+2696102T2+2520p3T3+p6T4 1 + 2520 T + 2696102 T^{2} + 2520 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−1686T+2089762T2−1686p3T3+p6T4 1 - 1686 T + 2089762 T^{2} - 1686 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−2100T+2861538T2−2100p3T3+p6T4 1 - 2100 T + 2861538 T^{2} - 2100 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

−8.881560446797156079801966532266, −8.738318990353735905656298600727, −7.890041417232435301536605048108, −7.81760025074929072264370962100, −7.31418063749322186238240207856, −7.16227694857730987845636546178, −6.27277263517035863369481384508, −6.05603364659633114838427010099, −5.72925217858204932895743878178, −5.53851373381353917707128931939, −4.72782190794944936840532925800, −4.50015044359038879456892995088, −3.74744236762429062186722687686, −3.65791206011254822425809517630, −2.77009233555504317668401669513, −2.50330456549303052910764198076, −2.02388374931356693390135508734, −1.97624827628271108411789854150, −0.947769038879111320623033531694, −0.60978641799910065497085969334, 0.60978641799910065497085969334, 0.947769038879111320623033531694, 1.97624827628271108411789854150, 2.02388374931356693390135508734, 2.50330456549303052910764198076, 2.77009233555504317668401669513, 3.65791206011254822425809517630, 3.74744236762429062186722687686, 4.50015044359038879456892995088, 4.72782190794944936840532925800, 5.53851373381353917707128931939, 5.72925217858204932895743878178, 6.05603364659633114838427010099, 6.27277263517035863369481384508, 7.16227694857730987845636546178, 7.31418063749322186238240207856, 7.81760025074929072264370962100, 7.890041417232435301536605048108, 8.738318990353735905656298600727, 8.881560446797156079801966532266

Graph of the ZZ-function along the critical line