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Properties

Label 2-350-5.4-c3-0-20
Degree 22
Conductor 350350
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 20.650620.6506
Root an. cond. 4.544304.54430
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s + 7i·7-s − 8i·8-s + 18·9-s − 17·11-s + 12i·12-s − 81i·13-s − 14·14-s + 16·16-s + 91i·17-s + 36i·18-s − 102·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s − 0.353i·8-s + 0.666·9-s − 0.465·11-s + 0.288i·12-s − 1.72i·13-s − 0.267·14-s + 0.250·16-s + 1.29i·17-s + 0.471i·18-s − 1.23·19-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+3/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 20.650620.6506
Root analytic conductor: 4.544304.54430
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ350(99,⋅)\chi_{350} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :3/2), 0.447+0.894i)(2,\ 350,\ (\ :3/2),\ 0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.3278991241.327899124
L(12)L(\frac12) ≈\approx 1.3278991241.327899124
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}
ppFp(T)F_p(T)
bad2 1−2iT 1 - 2iT
5 1 1
7 1−7iT 1 - 7iT
good3 1+3iT−27T2 1 + 3iT - 27T^{2}
11 1+17T+1.33e3T2 1 + 17T + 1.33e3T^{2}
13 1+81iT−2.19e3T2 1 + 81iT - 2.19e3T^{2}
17 1−91iT−4.91e3T2 1 - 91iT - 4.91e3T^{2}
19 1+102T+6.85e3T2 1 + 102T + 6.85e3T^{2}
23 1+90iT−1.21e4T2 1 + 90iT - 1.21e4T^{2}
29 1−129T+2.43e4T2 1 - 129T + 2.43e4T^{2}
31 1−116T+2.97e4T2 1 - 116T + 2.97e4T^{2}
37 1+314iT−5.06e4T2 1 + 314iT - 5.06e4T^{2}
41 1+124T+6.89e4T2 1 + 124T + 6.89e4T^{2}
43 1+434iT−7.95e4T2 1 + 434iT - 7.95e4T^{2}
47 1+497iT−1.03e5T2 1 + 497iT - 1.03e5T^{2}
53 1+584iT−1.48e5T2 1 + 584iT - 1.48e5T^{2}
59 1−332T+2.05e5T2 1 - 332T + 2.05e5T^{2}
61 1−220T+2.26e5T2 1 - 220T + 2.26e5T^{2}
67 1+384iT−3.00e5T2 1 + 384iT - 3.00e5T^{2}
71 1+664T+3.57e5T2 1 + 664T + 3.57e5T^{2}
73 1−230iT−3.89e5T2 1 - 230iT - 3.89e5T^{2}
79 1+361T+4.93e5T2 1 + 361T + 4.93e5T^{2}
83 1−1.17e3iT−5.71e5T2 1 - 1.17e3iT - 5.71e5T^{2}
89 1+40T+7.04e5T2 1 + 40T + 7.04e5T^{2}
97 1−175iT−9.12e5T2 1 - 175iT - 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.53733944730544680693454600320, −10.17854673533669318148578442753, −8.512137212124069104945325526275, −8.177000964108885115914933420210, −7.02864123599620133905285995885, −6.15215326550742097832947971474, −5.18887093482956099563357642590, −3.86648157111727698568618872245, −2.23422264810574348477291435214, −0.49097821067059405007372558968, 1.41548116007541528333977614715, 2.87608480210429984746953956274, 4.32249910981479415825078058819, 4.71695331950196216249684466903, 6.42831728315554137605693528732, 7.46075813183918820099287549608, 8.758281515806361199995495573082, 9.616651139737712384209593572268, 10.23074031385991962403494144818, 11.23060983851067093203376671653

Graph of the ZZ-function along the critical line