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Properties

Label 4-936e2-1.1-c3e2-0-1
Degree 44
Conductor 876096876096
Sign 11
Analytic cond. 3049.883049.88
Root an. cond. 7.431407.43140
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  + 4·5-s − 20·7-s + 60·11-s − 26·13-s + 84·17-s − 60·19-s + 72·23-s − 210·25-s + 124·29-s − 108·31-s − 80·35-s + 36·37-s + 52·41-s − 32·43-s + 428·47-s − 134·49-s − 380·53-s + 240·55-s + 1.42e3·59-s + 1.01e3·61-s − 104·65-s − 844·67-s + 868·71-s − 60·73-s − 1.20e3·77-s + 272·79-s + 1.25e3·83-s + ⋯
L(s)  = 1  + 0.357·5-s − 1.07·7-s + 1.64·11-s − 0.554·13-s + 1.19·17-s − 0.724·19-s + 0.652·23-s − 1.67·25-s + 0.794·29-s − 0.625·31-s − 0.386·35-s + 0.159·37-s + 0.198·41-s − 0.113·43-s + 1.32·47-s − 0.390·49-s − 0.984·53-s + 0.588·55-s + 3.13·59-s + 2.12·61-s − 0.198·65-s − 1.53·67-s + 1.45·71-s − 0.0961·73-s − 1.77·77-s + 0.387·79-s + 1.65·83-s + ⋯

Functional equation

Λ(s)=(876096s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(876096s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 876096876096    =    26⋅34⋅1322^{6} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 3049.883049.88
Root analytic conductor: 7.431407.43140
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 876096, ( :3/2,3/2), 1)(4,\ 876096,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.2267198493.226719849
L(12)L(\frac12) ≈\approx 3.2267198493.226719849
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
3 1 1
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1−4T+226T2−4p3T3+p6T4 1 - 4 T + 226 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+20T+534T2+20p3T3+p6T4 1 + 20 T + 534 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−60T+3114T2−60p3T3+p6T4 1 - 60 T + 3114 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−84T+10582T2−84p3T3+p6T4 1 - 84 T + 10582 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+60T+6526T2+60p3T3+p6T4 1 + 60 T + 6526 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−72T+14430T2−72p3T3+p6T4 1 - 72 T + 14430 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−124T−1586T2−124p3T3+p6T4 1 - 124 T - 1586 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+108T−16154T2+108p3T3+p6T4 1 + 108 T - 16154 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−36T+42382T2−36p3T3+p6T4 1 - 36 T + 42382 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−52T+76666T2−52p3T3+p6T4 1 - 52 T + 76666 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+32T+150198T2+32p3T3+p6T4 1 + 32 T + 150198 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−428T+116242T2−428p3T3+p6T4 1 - 428 T + 116242 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+380T+258142T2+380p3T3+p6T4 1 + 380 T + 258142 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−1420T+882490T2−1420p3T3+p6T4 1 - 1420 T + 882490 T^{2} - 1420 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−1012T+708206T2−1012p3T3+p6T4 1 - 1012 T + 708206 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+844T+778238T2+844p3T3+p6T4 1 + 844 T + 778238 T^{2} + 844 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−868T+888050T2−868p3T3+p6T4 1 - 868 T + 888050 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+60T+114886T2+60p3T3+p6T4 1 + 60 T + 114886 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−272T+993374T2−272p3T3+p6T4 1 - 272 T + 993374 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−1252T+1118698T2−1252p3T3+p6T4 1 - 1252 T + 1118698 T^{2} - 1252 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+572T+853306T2+572p3T3+p6T4 1 + 572 T + 853306 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−708T+1762390T2−708p3T3+p6T4 1 - 708 T + 1762390 T^{2} - 708 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

−9.682769275316558451439543479409, −9.665501630742496776000765918800, −8.989548486388880517970525139888, −8.945050999339779608466859535843, −8.026538617283859789435278816216, −7.986602093722320726290349830754, −7.11642283516373095862941225541, −6.93627993051916138822679883669, −6.42675915290428761164413041458, −6.15425401009738511870859127689, −5.43878472926334727798313626109, −5.37583526258642100078448071333, −4.31517351953994171765573131581, −4.15661282717541198967742916102, −3.35076686402295415440951550779, −3.30871327573630775011803171414, −2.23935713191443860958381587503, −1.96978322746292201857488326676, −0.980845958631757754395181003044, −0.55044611406476825340072769790, 0.55044611406476825340072769790, 0.980845958631757754395181003044, 1.96978322746292201857488326676, 2.23935713191443860958381587503, 3.30871327573630775011803171414, 3.35076686402295415440951550779, 4.15661282717541198967742916102, 4.31517351953994171765573131581, 5.37583526258642100078448071333, 5.43878472926334727798313626109, 6.15425401009738511870859127689, 6.42675915290428761164413041458, 6.93627993051916138822679883669, 7.11642283516373095862941225541, 7.986602093722320726290349830754, 8.026538617283859789435278816216, 8.945050999339779608466859535843, 8.989548486388880517970525139888, 9.665501630742496776000765918800, 9.682769275316558451439543479409

Graph of the ZZ-function along the critical line