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Properties

Label 4-1440e2-1.1-c3e2-0-2
Degree 44
Conductor 20736002073600
Sign 11
Analytic cond. 7218.667218.66
Root an. cond. 9.217529.21752
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  − 10·5-s − 8·7-s + 64·11-s + 12·13-s − 4·17-s − 208·19-s + 120·23-s + 75·25-s + 292·29-s − 176·31-s + 80·35-s − 356·37-s − 100·41-s + 376·43-s − 280·47-s − 38·49-s − 316·53-s − 640·55-s + 720·59-s − 1.26e3·61-s − 120·65-s + 744·67-s − 48·71-s − 940·73-s − 512·77-s − 32·79-s − 1.59e3·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.431·7-s + 1.75·11-s + 0.256·13-s − 0.0570·17-s − 2.51·19-s + 1.08·23-s + 3/5·25-s + 1.86·29-s − 1.01·31-s + 0.386·35-s − 1.58·37-s − 0.380·41-s + 1.33·43-s − 0.868·47-s − 0.110·49-s − 0.818·53-s − 1.56·55-s + 1.58·59-s − 2.66·61-s − 0.228·65-s + 1.35·67-s − 0.0802·71-s − 1.50·73-s − 0.757·77-s − 0.0455·79-s − 2.10·83-s + ⋯

Functional equation

Λ(s)=(2073600s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2073600s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20736002073600    =    210⋅34⋅522^{10} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 7218.667218.66
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2073600, ( :3/2,3/2), 1)(4,\ 2073600,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4046267441.404626744
L(12)L(\frac12) ≈\approx 1.4046267441.404626744
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
5C1C_1 (1+pT)2 ( 1 + p T )^{2}
good7D4D_{4} 1+8T+102T2+8p3T3+p6T4 1 + 8 T + 102 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−64T+2822T2−64p3T3+p6T4 1 - 64 T + 2822 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−12T+2894T2−12p3T3+p6T4 1 - 12 T + 2894 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+4T−3994T2+4p3T3+p6T4 1 + 4 T - 3994 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+208T+22998T2+208p3T3+p6T4 1 + 208 T + 22998 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−120T+25030T2−120p3T3+p6T4 1 - 120 T + 25030 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−292T+63950T2−292p3T3+p6T4 1 - 292 T + 63950 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+176T+66462T2+176p3T3+p6T4 1 + 176 T + 66462 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+356T+126846T2+356p3T3+p6T4 1 + 356 T + 126846 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+100T+101942T2+100p3T3+p6T4 1 + 100 T + 101942 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−376T+161502T2−376p3T3+p6T4 1 - 376 T + 161502 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+280T+223190T2+280p3T3+p6T4 1 + 280 T + 223190 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+316T+308894T2+316p3T3+p6T4 1 + 316 T + 308894 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−720T+530758T2−720p3T3+p6T4 1 - 720 T + 530758 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+1268T+831342T2+1268p3T3+p6T4 1 + 1268 T + 831342 T^{2} + 1268 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−744T+686894T2−744p3T3+p6T4 1 - 744 T + 686894 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+48T−424178T2+48p3T3+p6T4 1 + 48 T - 424178 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+940T+653334T2+940p3T3+p6T4 1 + 940 T + 653334 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+32T+276318T2+32p3T3+p6T4 1 + 32 T + 276318 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1592T+1724174T2+1592p3T3+p6T4 1 + 1592 T + 1724174 T^{2} + 1592 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−780T+1506742T2−780p3T3+p6T4 1 - 780 T + 1506742 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−1220T+2159046T2−1220p3T3+p6T4 1 - 1220 T + 2159046 T^{2} - 1220 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

−8.973801619874975540379668426171, −8.935767511904099718001182259529, −8.711478579753449418536952165435, −8.369270924290798597498347477850, −7.69974727351707514899423286269, −7.29102743881890859235277841509, −6.76799234624203607251051285552, −6.68204654785705396439806314942, −6.10331448735096360129436861705, −5.95382240961210189845530085716, −4.84696008836074368626689389313, −4.83175487950912518280131147525, −4.05064931601639028932095118005, −4.03307804656731597282123137115, −3.24952369985520529897544702800, −3.05372288131847205122141959279, −2.12912958084705137500288520384, −1.65180196271476136767296218214, −0.970065573736024977370780967225, −0.31248267891374571895479596989, 0.31248267891374571895479596989, 0.970065573736024977370780967225, 1.65180196271476136767296218214, 2.12912958084705137500288520384, 3.05372288131847205122141959279, 3.24952369985520529897544702800, 4.03307804656731597282123137115, 4.05064931601639028932095118005, 4.83175487950912518280131147525, 4.84696008836074368626689389313, 5.95382240961210189845530085716, 6.10331448735096360129436861705, 6.68204654785705396439806314942, 6.76799234624203607251051285552, 7.29102743881890859235277841509, 7.69974727351707514899423286269, 8.369270924290798597498347477850, 8.711478579753449418536952165435, 8.935767511904099718001182259529, 8.973801619874975540379668426171

Graph of the ZZ-function along the critical line