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Properties

Label 4-38e2-1.1-c3e2-0-3
Degree 44
Conductor 14441444
Sign 11
Analytic cond. 5.026885.02688
Root an. cond. 1.497351.49735
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  + 4·2-s + 9·3-s + 12·4-s − 9·5-s + 36·6-s − 18·7-s + 32·8-s + 25·9-s − 36·10-s − 17·11-s + 108·12-s + 17·13-s − 72·14-s − 81·15-s + 80·16-s − 80·17-s + 100·18-s + 38·19-s − 108·20-s − 162·21-s − 68·22-s + 73·23-s + 288·24-s − 25·25-s + 68·26-s − 36·27-s − 216·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.804·5-s + 2.44·6-s − 0.971·7-s + 1.41·8-s + 0.925·9-s − 1.13·10-s − 0.465·11-s + 2.59·12-s + 0.362·13-s − 1.37·14-s − 1.39·15-s + 5/4·16-s − 1.14·17-s + 1.30·18-s + 0.458·19-s − 1.20·20-s − 1.68·21-s − 0.658·22-s + 0.661·23-s + 2.44·24-s − 1/5·25-s + 0.512·26-s − 0.256·27-s − 1.45·28-s + ⋯

Functional equation

Λ(s)=(1444s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1444s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14441444    =    22⋅1922^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 5.026885.02688
Root analytic conductor: 1.497351.49735
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1444, ( :3/2,3/2), 1)(4,\ 1444,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.3035288324.303528832
L(12)L(\frac12) ≈\approx 4.3035288324.303528832
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)
bad2C1C_1 (1−pT)2 ( 1 - p T )^{2}
19C1C_1 (1−pT)2 ( 1 - p T )^{2}
good3D4D_{4} 1−p2T+56T2−p5T3+p6T4 1 - p^{2} T + 56 T^{2} - p^{5} T^{3} + p^{6} T^{4}
5D4D_{4} 1+9T+106T2+9p3T3+p6T4 1 + 9 T + 106 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+18T+475T2+18p3T3+p6T4 1 + 18 T + 475 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+17T+2716T2+17p3T3+p6T4 1 + 17 T + 2716 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−17T+1382T2−17p3T3+p6T4 1 - 17 T + 1382 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+80T+11353T2+80p3T3+p6T4 1 + 80 T + 11353 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−73T+22582T2−73p3T3+p6T4 1 - 73 T + 22582 T^{2} - 73 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−3T+40732T2−3p3T3+p6T4 1 - 3 T + 40732 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−212T+35486T2−212p3T3+p6T4 1 - 212 T + 35486 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−192T+96214T2−192p3T3+p6T4 1 - 192 T + 96214 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+50T+136642T2+50p3T3+p6T4 1 + 50 T + 136642 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−677T+272702T2−677p3T3+p6T4 1 - 677 T + 272702 T^{2} - 677 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+389T+153478T2+389p3T3+p6T4 1 + 389 T + 153478 T^{2} + 389 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+23pT+663970T2+23p4T3+p6T4 1 + 23 p T + 663970 T^{2} + 23 p^{4} T^{3} + p^{6} T^{4}
59D4D_{4} 1+287T+419944T2+287p3T3+p6T4 1 + 287 T + 419944 T^{2} + 287 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−313T+253596T2−313p3T3+p6T4 1 - 313 T + 253596 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−1223T+867254T2−1223p3T3+p6T4 1 - 1223 T + 867254 T^{2} - 1223 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−200T+480250T2−200p3T3+p6T4 1 - 200 T + 480250 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−378T+195883T2−378p3T3+p6T4 1 - 378 T + 195883 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−1350T+1380310T2−1350p3T3+p6T4 1 - 1350 T + 1380310 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+670T+568942T2+670p3T3+p6T4 1 + 670 T + 568942 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+236T+778834T2+236p3T3+p6T4 1 + 236 T + 778834 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−1294T+2054082T2−1294p3T3+p6T4 1 - 1294 T + 2054082 T^{2} - 1294 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

−15.72995924244702178401287918599, −15.58716924581823310625876813870, −14.71090850537627076555606662756, −14.38500981277940223423798464016, −13.68656564713630820883903045839, −13.38775329037008038079963478557, −12.72576187683773906907546760818, −12.30313414350694990763024205308, −11.12894287555751293338822293772, −11.08453363412813674368454222799, −9.697880035208362787704356901554, −9.261357644577212813933289374130, −8.084959882271028323143027320107, −7.996750157870826228222825275427, −6.83945744702810489928646858932, −6.20969467878938420561383725721, −4.89012081848617119485394458011, −3.93149329831377357804181671820, −3.16303426781052515390090952770, −2.55507133412447475098459378132, 2.55507133412447475098459378132, 3.16303426781052515390090952770, 3.93149329831377357804181671820, 4.89012081848617119485394458011, 6.20969467878938420561383725721, 6.83945744702810489928646858932, 7.996750157870826228222825275427, 8.084959882271028323143027320107, 9.261357644577212813933289374130, 9.697880035208362787704356901554, 11.08453363412813674368454222799, 11.12894287555751293338822293772, 12.30313414350694990763024205308, 12.72576187683773906907546760818, 13.38775329037008038079963478557, 13.68656564713630820883903045839, 14.38500981277940223423798464016, 14.71090850537627076555606662756, 15.58716924581823310625876813870, 15.72995924244702178401287918599

Graph of the ZZ-function along the critical line