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Properties

Label 4-490e2-1.1-c5e2-0-0
Degree 44
Conductor 240100240100
Sign 11
Analytic cond. 6176.086176.08
Root an. cond. 8.864998.86499
Motivic weight 55
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  − 8·2-s + 20·3-s + 48·4-s − 50·5-s − 160·6-s − 256·8-s + 151·9-s + 400·10-s − 860·11-s + 960·12-s − 1.27e3·13-s − 1.00e3·15-s + 1.28e3·16-s + 250·17-s − 1.20e3·18-s + 460·19-s − 2.40e3·20-s + 6.88e3·22-s − 2.46e3·23-s − 5.12e3·24-s + 1.87e3·25-s + 1.01e4·26-s + 2.90e3·27-s + 682·29-s + 8.00e3·30-s − 5.74e3·31-s − 6.14e3·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.28·3-s + 3/2·4-s − 0.894·5-s − 1.81·6-s − 1.41·8-s + 0.621·9-s + 1.26·10-s − 2.14·11-s + 1.92·12-s − 2.08·13-s − 1.14·15-s + 5/4·16-s + 0.209·17-s − 0.878·18-s + 0.292·19-s − 1.34·20-s + 3.03·22-s − 0.972·23-s − 1.81·24-s + 3/5·25-s + 2.94·26-s + 0.765·27-s + 0.150·29-s + 1.62·30-s − 1.07·31-s − 1.06·32-s + ⋯

Functional equation

Λ(s)=(240100s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(240100s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 240100240100    =    22⋅52⋅742^{2} \cdot 5^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 6176.086176.08
Root analytic conductor: 8.864998.86499
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 240100, ( :5/2,5/2), 1)(4,\ 240100,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.35393308340.3539330834
L(12)L(\frac12) ≈\approx 0.35393308340.3539330834
L(72)L(\frac{7}{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+p2T)2 ( 1 + p^{2} T )^{2}
5C1C_1 (1+p2T)2 ( 1 + p^{2} T )^{2}
7 1 1
good3D4D_{4} 1−20T+83pT2−20p5T3+p10T4 1 - 20 T + 83 p T^{2} - 20 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1+860T+498577T2+860p5T3+p10T4 1 + 860 T + 498577 T^{2} + 860 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+1270T+982703T2+1270p5T3+p10T4 1 + 1270 T + 982703 T^{2} + 1270 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1−250T+908827T2−250p5T3+p10T4 1 - 250 T + 908827 T^{2} - 250 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1−460T+3446810T2−460p5T3+p10T4 1 - 460 T + 3446810 T^{2} - 460 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+2468T+14361742T2+2468p5T3+p10T4 1 + 2468 T + 14361742 T^{2} + 2468 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1−682T+40296079T2−682p5T3+p10T4 1 - 682 T + 40296079 T^{2} - 682 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1+5740T+62389410T2+5740p5T3+p10T4 1 + 5740 T + 62389410 T^{2} + 5740 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1−2532T+14892970T2−2532p5T3+p10T4 1 - 2532 T + 14892970 T^{2} - 2532 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1−22560T+333650190T2−22560p5T3+p10T4 1 - 22560 T + 333650190 T^{2} - 22560 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1−6484T+77928650T2−6484p5T3+p10T4 1 - 6484 T + 77928650 T^{2} - 6484 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1−1840T+456985661T2−1840p5T3+p10T4 1 - 1840 T + 456985661 T^{2} - 1840 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1−26508T+493888302T2−26508p5T3+p10T4 1 - 26508 T + 493888302 T^{2} - 26508 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1−35520T+1379363730T2−35520p5T3+p10T4 1 - 35520 T + 1379363730 T^{2} - 35520 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+68220T+2580926554T2+68220p5T3+p10T4 1 + 68220 T + 2580926554 T^{2} + 68220 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+22640T+2718901314T2+22640p5T3+p10T4 1 + 22640 T + 2718901314 T^{2} + 22640 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+33928T+2679665998T2+33928p5T3+p10T4 1 + 33928 T + 2679665998 T^{2} + 33928 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1+18140T+4108700294T2+18140p5T3+p10T4 1 + 18140 T + 4108700294 T^{2} + 18140 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1−23600T+475376373T2−23600p5T3+p10T4 1 - 23600 T + 475376373 T^{2} - 23600 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1−67840T+5922418934T2−67840p5T3+p10T4 1 - 67840 T + 5922418934 T^{2} - 67840 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1+72380T+12375892498T2+72380p5T3+p10T4 1 + 72380 T + 12375892498 T^{2} + 72380 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1−143650T+22148969939T2−143650p5T3+p10T4 1 - 143650 T + 22148969939 T^{2} - 143650 p^{5} T^{3} + p^{10} 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

−10.39292295567185967646553741417, −9.789295908146316657999734098261, −9.382001365891373833408912323967, −9.152985914903881167227496881989, −8.418520252542477435284001295318, −8.178936550783361417698883119005, −7.65144877939155309998701134570, −7.54303957321425461872275390644, −7.29323367223223959680224458252, −6.57223000940365173756931568105, −5.61013865495016099837504495520, −5.40239927517696686846148352877, −4.54708924608472828735729735832, −4.08882336135706820048642635112, −3.08451726578986814065936231090, −2.82410805426202497546245596384, −2.42211639003030909002763651877, −1.94142255598356978060036056076, −0.835814258779458793626324416181, −0.19790564121808778296996891475, 0.19790564121808778296996891475, 0.835814258779458793626324416181, 1.94142255598356978060036056076, 2.42211639003030909002763651877, 2.82410805426202497546245596384, 3.08451726578986814065936231090, 4.08882336135706820048642635112, 4.54708924608472828735729735832, 5.40239927517696686846148352877, 5.61013865495016099837504495520, 6.57223000940365173756931568105, 7.29323367223223959680224458252, 7.54303957321425461872275390644, 7.65144877939155309998701134570, 8.178936550783361417698883119005, 8.418520252542477435284001295318, 9.152985914903881167227496881989, 9.382001365891373833408912323967, 9.789295908146316657999734098261, 10.39292295567185967646553741417

Graph of the ZZ-function along the critical line