↑

Properties

Label 2-350-5.4-c3-0-15
Degree 22
Conductor 350350
Sign 0.894−0.447i0.894 - 0.447i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 4i·3-s − 4·4-s − 8·6-s − 7i·7-s − 8i·8-s + 11·9-s + 5·11-s − 16i·12-s − 82i·13-s + 14·14-s + 16·16-s − 12i·17-s + 22i·18-s + 42·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.769i·3-s − 0.5·4-s − 0.544·6-s − 0.377i·7-s − 0.353i·8-s + 0.407·9-s + 0.137·11-s − 0.384i·12-s − 1.74i·13-s + 0.267·14-s + 0.250·16-s − 0.171i·17-s + 0.288i·18-s + 0.507·19-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.894−0.447i0.894 - 0.447i
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), 0.894−0.447i)(2,\ 350,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.7543148121.754314812
L(12)L(\frac12) ≈\approx 1.7543148121.754314812
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 1−4iT−27T2 1 - 4iT - 27T^{2}
11 1−5T+1.33e3T2 1 - 5T + 1.33e3T^{2}
13 1+82iT−2.19e3T2 1 + 82iT - 2.19e3T^{2}
17 1+12iT−4.91e3T2 1 + 12iT - 4.91e3T^{2}
19 1−42T+6.85e3T2 1 - 42T + 6.85e3T^{2}
23 1+175iT−1.21e4T2 1 + 175iT - 1.21e4T^{2}
29 1+T+2.43e4T2 1 + T + 2.43e4T^{2}
31 1−226T+2.97e4T2 1 - 226T + 2.97e4T^{2}
37 1+19iT−5.06e4T2 1 + 19iT - 5.06e4T^{2}
41 1−16T+6.89e4T2 1 - 16T + 6.89e4T^{2}
43 1+281iT−7.95e4T2 1 + 281iT - 7.95e4T^{2}
47 1−334iT−1.03e5T2 1 - 334iT - 1.03e5T^{2}
53 1−398iT−1.48e5T2 1 - 398iT - 1.48e5T^{2}
59 1+106T+2.05e5T2 1 + 106T + 2.05e5T^{2}
61 1−48T+2.26e5T2 1 - 48T + 2.26e5T^{2}
67 1−483iT−3.00e5T2 1 - 483iT - 3.00e5T^{2}
71 1+15T+3.57e5T2 1 + 15T + 3.57e5T^{2}
73 1+1.04e3iT−3.89e5T2 1 + 1.04e3iT - 3.89e5T^{2}
79 1−1.25e3T+4.93e5T2 1 - 1.25e3T + 4.93e5T^{2}
83 1+758iT−5.71e5T2 1 + 758iT - 5.71e5T^{2}
89 1+86T+7.04e5T2 1 + 86T + 7.04e5T^{2}
97 1−710iT−9.12e5T2 1 - 710iT - 9.12e5T^{2}
show more
show less
   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

−10.66560709921042268840084654607, −10.26925474773874803237329613024, −9.290509715300972407834825053010, −8.228567947035707006389265355201, −7.39117894098126002599413150119, −6.25705035058007297310911286034, −5.13235829345022881097627823780, −4.28160841356787353520963141142, −3.05792702224212663854573183874, −0.72845727608546573999796201431, 1.24784989889817536008594891379, 2.18692095830498861440865461940, 3.71589610073648110915943104635, 4.86745854549275649088650486958, 6.26373415691182703731166237286, 7.16949694773755817967264224995, 8.232379886272797760183249487025, 9.330204703659582403276758708952, 9.948296661772883470218523104855, 11.38989181369026411067706938860

Graph of the ZZ-function along the critical line