↑

Properties

Label 4-525e2-1.1-c3e2-0-2
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 959.512959.512
Root an. cond. 5.565605.56560
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9·4-s − 9·9-s + 24·11-s + 17·16-s + 184·19-s + 116·29-s − 448·31-s + 81·36-s + 36·41-s − 216·44-s − 49·49-s − 760·59-s + 1.43e3·61-s + 423·64-s − 1.92e3·71-s − 1.65e3·76-s − 1.79e3·79-s + 81·81-s + 2.07e3·89-s − 216·99-s + 92·101-s + 756·109-s − 1.04e3·116-s − 2.23e3·121-s + 4.03e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 9/8·4-s − 1/3·9-s + 0.657·11-s + 0.265·16-s + 2.22·19-s + 0.742·29-s − 2.59·31-s + 3/8·36-s + 0.137·41-s − 0.740·44-s − 1/7·49-s − 1.67·59-s + 3.01·61-s + 0.826·64-s − 3.20·71-s − 2.49·76-s − 2.55·79-s + 1/9·81-s + 2.47·89-s − 0.219·99-s + 0.0906·101-s + 0.664·109-s − 0.835·116-s − 1.67·121-s + 2.92·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 959.512959.512
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 275625, ( :3/2,3/2), 1)(4,\ 275625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.3612551161.361255116
L(12)L(\frac12) ≈\approx 1.3612551161.361255116
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)
bad3C2C_2 1+p2T2 1 + p^{2} T^{2}
5 1 1
7C2C_2 1+p2T2 1 + p^{2} T^{2}
good2C22C_2^2 1+9T2+p6T4 1 + 9 T^{2} + p^{6} T^{4}
11C2C_2 (1−12T+p3T2)2 ( 1 - 12 T + p^{3} T^{2} )^{2}
13C22C_2^2 1−3494T2+p6T4 1 - 3494 T^{2} + p^{6} T^{4}
17C22C_2^2 1+8130T2+p6T4 1 + 8130 T^{2} + p^{6} T^{4}
19C2C_2 (1−92T+p3T2)2 ( 1 - 92 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−11790T2+p6T4 1 - 11790 T^{2} + p^{6} T^{4}
29C2C_2 (1−2pT+p3T2)2 ( 1 - 2 p T + p^{3} T^{2} )^{2}
31C2C_2 (1+224T+p3T2)2 ( 1 + 224 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−79990T2+p6T4 1 - 79990 T^{2} + p^{6} T^{4}
41C2C_2 (1−18T+p3T2)2 ( 1 - 18 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−43414T2+p6T4 1 - 43414 T^{2} + p^{6} T^{4}
47C22C_2^2 1−164382T2+p6T4 1 - 164382 T^{2} + p^{6} T^{4}
53C22C_2^2 1+270762T2+p6T4 1 + 270762 T^{2} + p^{6} T^{4}
59C2C_2 (1+380T+p3T2)2 ( 1 + 380 T + p^{3} T^{2} )^{2}
61C2C_2 (1−718T+p3T2)2 ( 1 - 718 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−431782T2+p6T4 1 - 431782 T^{2} + p^{6} T^{4}
71C2C_2 (1+960T+p3T2)2 ( 1 + 960 T + p^{3} T^{2} )^{2}
73C22C_2^2 1+358322T2+p6T4 1 + 358322 T^{2} + p^{6} T^{4}
79C2C_2 (1+896T+p3T2)2 ( 1 + 896 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−953478T2+p6T4 1 - 953478 T^{2} + p^{6} T^{4}
89C2C_2 (1−1038T+p3T2)2 ( 1 - 1038 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1332542T2+p6T4 1 - 1332542 T^{2} + p^{6} T^{4}
show more
show less
   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

−10.42595755866739323090356494508, −10.25933895455305968310546091819, −9.674178732454685000049996961408, −9.227145245004230714359924725357, −8.887350438553412334698872153535, −8.809394589780622704061818389405, −7.85908873739547548724140952246, −7.66615696543583308656028630782, −7.08702854628003906995801771606, −6.63258564425627448251898906408, −5.88921960399178111643206010912, −5.34578897225955341319423305355, −5.24910831615715771093284791001, −4.38375148537717703945141083609, −4.04907078036679444790608634026, −3.32461736289031074119610053593, −2.97792479975793590736842334955, −1.89776166478418392284370755303, −1.18480633670555135579211835896, −0.40926267412199916161492296753, 0.40926267412199916161492296753, 1.18480633670555135579211835896, 1.89776166478418392284370755303, 2.97792479975793590736842334955, 3.32461736289031074119610053593, 4.04907078036679444790608634026, 4.38375148537717703945141083609, 5.24910831615715771093284791001, 5.34578897225955341319423305355, 5.88921960399178111643206010912, 6.63258564425627448251898906408, 7.08702854628003906995801771606, 7.66615696543583308656028630782, 7.85908873739547548724140952246, 8.809394589780622704061818389405, 8.887350438553412334698872153535, 9.227145245004230714359924725357, 9.674178732454685000049996961408, 10.25933895455305968310546091819, 10.42595755866739323090356494508

Graph of the ZZ-function along the critical line