↑

Properties

Label 4-350e2-1.1-c3e2-0-4
Degree 44
Conductor 122500122500
Sign 11
Analytic cond. 426.450426.450
Root an. cond. 4.544304.54430
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  − 2·2-s − 3-s + 2·6-s + 20·7-s + 8·8-s + 27·9-s − 35·11-s − 132·13-s − 40·14-s − 16·16-s + 59·17-s − 54·18-s − 137·19-s − 20·21-s + 70·22-s − 7·23-s − 8·24-s + 264·26-s − 80·27-s + 212·29-s − 75·31-s + 35·33-s − 118·34-s + 11·37-s + 274·38-s + 132·39-s − 996·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.192·3-s + 0.136·6-s + 1.07·7-s + 0.353·8-s + 9-s − 0.959·11-s − 2.81·13-s − 0.763·14-s − 1/4·16-s + 0.841·17-s − 0.707·18-s − 1.65·19-s − 0.207·21-s + 0.678·22-s − 0.0634·23-s − 0.0680·24-s + 1.99·26-s − 0.570·27-s + 1.35·29-s − 0.434·31-s + 0.184·33-s − 0.595·34-s + 0.0488·37-s + 1.16·38-s + 0.541·39-s − 3.79·41-s + ⋯

Functional equation

Λ(s)=(122500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(122500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 122500122500    =    22⋅54⋅722^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 426.450426.450
Root analytic conductor: 4.544304.54430
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 122500, ( :3/2,3/2), 1)(4,\ 122500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.14765747650.1476574765
L(12)L(\frac12) ≈\approx 0.14765747650.1476574765
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)
bad2C2C_2 1+pT+p2T2 1 + p T + p^{2} T^{2}
5 1 1
7C2C_2 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
good3C22C_2^2 1+T−26T2+p3T3+p6T4 1 + T - 26 T^{2} + p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1+35T−106T2+35p3T3+p6T4 1 + 35 T - 106 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4}
13C2C_2 (1+66T+p3T2)2 ( 1 + 66 T + p^{3} T^{2} )^{2}
17C22C_2^2 1−59T−1432T2−59p3T3+p6T4 1 - 59 T - 1432 T^{2} - 59 p^{3} T^{3} + p^{6} T^{4}
19C22C_2^2 1+137T+11910T2+137p3T3+p6T4 1 + 137 T + 11910 T^{2} + 137 p^{3} T^{3} + p^{6} T^{4}
23C22C_2^2 1+7T−12118T2+7p3T3+p6T4 1 + 7 T - 12118 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4}
29C2C_2 (1−106T+p3T2)2 ( 1 - 106 T + p^{3} T^{2} )^{2}
31C22C_2^2 1+75T−24166T2+75p3T3+p6T4 1 + 75 T - 24166 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4}
37C22C_2^2 1−11T−50532T2−11p3T3+p6T4 1 - 11 T - 50532 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4}
41C2C_2 (1+498T+p3T2)2 ( 1 + 498 T + p^{3} T^{2} )^{2}
43C2C_2 (1+260T+p3T2)2 ( 1 + 260 T + p^{3} T^{2} )^{2}
47C22C_2^2 1+171T−74582T2+171p3T3+p6T4 1 + 171 T - 74582 T^{2} + 171 p^{3} T^{3} + p^{6} T^{4}
53C22C_2^2 1+417T+25012T2+417p3T3+p6T4 1 + 417 T + 25012 T^{2} + 417 p^{3} T^{3} + p^{6} T^{4}
59C22C_2^2 1−17T−205090T2−17p3T3+p6T4 1 - 17 T - 205090 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1+51T−224380T2+51p3T3+p6T4 1 + 51 T - 224380 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1−439T−108042T2−439p3T3+p6T4 1 - 439 T - 108042 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1+784T+p3T2)2 ( 1 + 784 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−295T−301992T2−295p3T3+p6T4 1 - 295 T - 301992 T^{2} - 295 p^{3} T^{3} + p^{6} T^{4}
79C22C_2^2 1−495T−248014T2−495p3T3+p6T4 1 - 495 T - 248014 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4}
83C2C_2 (1+932T+p3T2)2 ( 1 + 932 T + p^{3} T^{2} )^{2}
89C22C_2^2 1−873T+57160T2−873p3T3+p6T4 1 - 873 T + 57160 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4}
97C2C_2 (1−290T+p3T2)2 ( 1 - 290 T + p^{3} T^{2} )^{2}
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

−11.44499634760613249600918936028, −10.41816610053329578691005083906, −10.27313561511752907770263618492, −10.00372359237019322727147659437, −9.780593995027672180449217878621, −8.782262332110583015717920623212, −8.530726450427554341367043069230, −7.84131566100644516096246478486, −7.77771793735576789885569481978, −6.93693587096006716564788222343, −6.92218878783427357446642331818, −5.96494565749609450939243639067, −4.96023506311919722308434974509, −4.81390782737162006274418269089, −4.77837759845928790766213766283, −3.60052821760592070351233374361, −2.75248657316652977619590646714, −1.92961064648979968451739817522, −1.58885509511434552720671401737, −0.14923568357506035820431349329, 0.14923568357506035820431349329, 1.58885509511434552720671401737, 1.92961064648979968451739817522, 2.75248657316652977619590646714, 3.60052821760592070351233374361, 4.77837759845928790766213766283, 4.81390782737162006274418269089, 4.96023506311919722308434974509, 5.96494565749609450939243639067, 6.92218878783427357446642331818, 6.93693587096006716564788222343, 7.77771793735576789885569481978, 7.84131566100644516096246478486, 8.530726450427554341367043069230, 8.782262332110583015717920623212, 9.780593995027672180449217878621, 10.00372359237019322727147659437, 10.27313561511752907770263618492, 10.41816610053329578691005083906, 11.44499634760613249600918936028

Graph of the ZZ-function along the critical line