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Properties

Label 4-480e2-1.1-c3e2-0-1
Degree 44
Conductor 230400230400
Sign 11
Analytic cond. 802.074802.074
Root an. cond. 5.321745.32174
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  − 52·7-s − 9·9-s + 12·17-s + 296·23-s − 25·25-s + 196·31-s + 644·41-s − 952·47-s + 1.34e3·49-s + 468·63-s + 1.49e3·71-s + 2.32e3·73-s − 620·79-s + 81·81-s − 980·89-s + 2.33e3·97-s + 2.11e3·103-s − 3.51e3·113-s − 624·119-s + 1.06e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 108·153-s + ⋯
L(s)  = 1  − 2.80·7-s − 1/3·9-s + 0.171·17-s + 2.68·23-s − 1/5·25-s + 1.13·31-s + 2.45·41-s − 2.95·47-s + 3.91·49-s + 0.935·63-s + 2.50·71-s + 3.72·73-s − 0.882·79-s + 1/9·81-s − 1.16·89-s + 2.44·97-s + 2.02·103-s − 2.92·113-s − 0.480·119-s + 0.797·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.0570·153-s + ⋯

Functional equation

Λ(s)=(230400s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(230400s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 230400230400    =    210⋅32⋅522^{10} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 802.074802.074
Root analytic conductor: 5.321745.32174
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 230400, ( :3/2,3/2), 1)(4,\ 230400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4421635051.442163505
L(12)L(\frac12) ≈\approx 1.4421635051.442163505
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
3C2C_2 1+p2T2 1 + p^{2} T^{2}
5C2C_2 1+p2T2 1 + p^{2} T^{2}
good7C2C_2 (1+26T+p3T2)2 ( 1 + 26 T + p^{3} T^{2} )^{2}
11C22C_2^2 1−1062T2+p6T4 1 - 1062 T^{2} + p^{6} T^{4}
13C22C_2^2 1−4250T2+p6T4 1 - 4250 T^{2} + p^{6} T^{4}
17C2C_2 (1−6T+p3T2)2 ( 1 - 6 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−13702T2+p6T4 1 - 13702 T^{2} + p^{6} T^{4}
23C2C_2 (1−148T+p3T2)2 ( 1 - 148 T + p^{3} T^{2} )^{2}
29C22C_2^2 1+37658T2+p6T4 1 + 37658 T^{2} + p^{6} T^{4}
31C2C_2 (1−98T+p3T2)2 ( 1 - 98 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−8890T2+p6T4 1 - 8890 T^{2} + p^{6} T^{4}
41C2C_2 (1−322T+p3T2)2 ( 1 - 322 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−8470T2+p6T4 1 - 8470 T^{2} + p^{6} T^{4}
47C2C_2 (1+476T+p3T2)2 ( 1 + 476 T + p^{3} T^{2} )^{2}
53C22C_2^2 1−283830T2+p6T4 1 - 283830 T^{2} + p^{6} T^{4}
59C22C_2^2 1−408822T2+p6T4 1 - 408822 T^{2} + p^{6} T^{4}
61C22C_2^2 1−447562T2+p6T4 1 - 447562 T^{2} + p^{6} T^{4}
67C22C_2^2 1−594470T2+p6T4 1 - 594470 T^{2} + p^{6} T^{4}
71C2C_2 (1−748T+p3T2)2 ( 1 - 748 T + p^{3} T^{2} )^{2}
73C2C_2 (1−1162T+p3T2)2 ( 1 - 1162 T + p^{3} T^{2} )^{2}
79C2C_2 (1+310T+p3T2)2 ( 1 + 310 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−127510T2+p6T4 1 - 127510 T^{2} + p^{6} T^{4}
89C2C_2 (1+490T+p3T2)2 ( 1 + 490 T + p^{3} T^{2} )^{2}
97C2C_2 (1−1166T+p3T2)2 ( 1 - 1166 T + p^{3} T^{2} )^{2}
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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

−10.81221146958276890219999037950, −10.19550015558394138050066903886, −9.731264922067917154291395353671, −9.632961744282965074547330146565, −8.973889891101352742010430222744, −8.941459008143920262295456266787, −7.930506007637125009815533984774, −7.73031010220982061576386932080, −6.71704685867300701918069384189, −6.67248542525218255007456043378, −6.44969365685989810280548378669, −5.72609929205555750462184821079, −5.15596519191607597264847631728, −4.62794530931952565132260440889, −3.63831042779824770018486137225, −3.43114311150400801973272458921, −2.84302535743211056896166605190, −2.42948734984238144700471620260, −1.00756375656053006301426759580, −0.46254395905598964464502678398, 0.46254395905598964464502678398, 1.00756375656053006301426759580, 2.42948734984238144700471620260, 2.84302535743211056896166605190, 3.43114311150400801973272458921, 3.63831042779824770018486137225, 4.62794530931952565132260440889, 5.15596519191607597264847631728, 5.72609929205555750462184821079, 6.44969365685989810280548378669, 6.67248542525218255007456043378, 6.71704685867300701918069384189, 7.73031010220982061576386932080, 7.930506007637125009815533984774, 8.941459008143920262295456266787, 8.973889891101352742010430222744, 9.632961744282965074547330146565, 9.731264922067917154291395353671, 10.19550015558394138050066903886, 10.81221146958276890219999037950

Graph of the ZZ-function along the critical line