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Properties

Label 4-1440e2-1.1-c3e2-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·5-s − 60·11-s − 24·19-s + 275·25-s − 128·29-s − 624·31-s + 748·41-s + 362·49-s − 1.20e3·55-s + 1.02e3·59-s + 1.50e3·61-s + 1.84e3·71-s − 144·79-s − 580·89-s − 480·95-s − 208·101-s − 1.14e3·109-s + 38·121-s + 3.00e3·125-s + 127-s + 131-s + 137-s + 139-s − 2.56e3·145-s + 149-s + 151-s − 1.24e4·155-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.64·11-s − 0.289·19-s + 11/5·25-s − 0.819·29-s − 3.61·31-s + 2.84·41-s + 1.05·49-s − 2.94·55-s + 2.25·59-s + 3.16·61-s + 3.08·71-s − 0.205·79-s − 0.690·89-s − 0.518·95-s − 0.204·101-s − 1.00·109-s + 0.0285·121-s + 2.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 1.46·145-s + 0.000549·149-s + 0.000538·151-s − 6.46·155-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 3.7059759553.705975955
L(12)L(\frac12) ≈\approx 3.7059759553.705975955
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
5C2C_2 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
good7C22C_2^2 1−362T2+p6T4 1 - 362 T^{2} + p^{6} T^{4}
11C2C_2 (1+30T+p3T2)2 ( 1 + 30 T + p^{3} T^{2} )^{2}
13C22C_2^2 1−2950T2+p6T4 1 - 2950 T^{2} + p^{6} T^{4}
17C22C_2^2 1−4926T2+p6T4 1 - 4926 T^{2} + p^{6} T^{4}
19C2C_2 (1+12T+p3T2)2 ( 1 + 12 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−19150T2+p6T4 1 - 19150 T^{2} + p^{6} T^{4}
29C2C_2 (1+64T+p3T2)2 ( 1 + 64 T + p^{3} T^{2} )^{2}
31C2C_2 (1+312T+p3T2)2 ( 1 + 312 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−82262T2+p6T4 1 - 82262 T^{2} + p^{6} T^{4}
41C2C_2 (1−374T+p3T2)2 ( 1 - 374 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+60010T2+p6T4 1 + 60010 T^{2} + p^{6} T^{4}
47C22C_2^2 1−190222T2+p6T4 1 - 190222 T^{2} + p^{6} T^{4}
53C22C_2^2 1−98838T2+p6T4 1 - 98838 T^{2} + p^{6} T^{4}
59C2C_2 (1−510T+p3T2)2 ( 1 - 510 T + p^{3} T^{2} )^{2}
61C2C_2 (1−754T+p3T2)2 ( 1 - 754 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−454070T2+p6T4 1 - 454070 T^{2} + p^{6} T^{4}
71C2C_2 (1−924T+p3T2)2 ( 1 - 924 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−662434T2+p6T4 1 - 662434 T^{2} + p^{6} T^{4}
79C2C_2 (1+72T+p3T2)2 ( 1 + 72 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−1119238T2+p6T4 1 - 1119238 T^{2} + p^{6} T^{4}
89C2C_2 (1+290T+p3T2)2 ( 1 + 290 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1683970T2+p6T4 1 - 1683970 T^{2} + 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

−9.541742100117204704659214908722, −9.139510755152034486241557464282, −8.436960427712506281639273600523, −8.419758107668550265432740129874, −7.60274664120605105313852704791, −7.36052496136214643269450979987, −6.96264074401133518583514882226, −6.50482752304830160907328819271, −5.88574156159852084089795592708, −5.44376429563268052804529886726, −5.37802717042391699557181664807, −5.20915532550212545101821080641, −4.12519231879542606167585001214, −3.92669585153679174060811840533, −3.20567447041666559858191237600, −2.48062156862639189626744594477, −2.14735511111703054915887147893, −2.03941806773108514571708023956, −1.01241873053732260153419952593, −0.46209766901744422265592530698, 0.46209766901744422265592530698, 1.01241873053732260153419952593, 2.03941806773108514571708023956, 2.14735511111703054915887147893, 2.48062156862639189626744594477, 3.20567447041666559858191237600, 3.92669585153679174060811840533, 4.12519231879542606167585001214, 5.20915532550212545101821080641, 5.37802717042391699557181664807, 5.44376429563268052804529886726, 5.88574156159852084089795592708, 6.50482752304830160907328819271, 6.96264074401133518583514882226, 7.36052496136214643269450979987, 7.60274664120605105313852704791, 8.419758107668550265432740129874, 8.436960427712506281639273600523, 9.139510755152034486241557464282, 9.541742100117204704659214908722

Graph of the ZZ-function along the critical line