Properties

Label 2-1445-5.4-c1-0-112
Degree 22
Conductor 14451445
Sign 0.07960.996i-0.0796 - 0.996i
Analytic cond. 11.538311.5383
Root an. cond. 3.396813.39681
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80i·2-s − 0.561i·3-s − 1.25·4-s + (−0.178 − 2.22i)5-s − 1.01·6-s + 3.11i·7-s − 1.34i·8-s + 2.68·9-s + (−4.01 + 0.321i)10-s − 5.61·11-s + 0.703i·12-s + 1.24i·13-s + 5.61·14-s + (−1.25 + 0.100i)15-s − 4.93·16-s + ⋯
L(s)  = 1  − 1.27i·2-s − 0.324i·3-s − 0.626·4-s + (−0.0796 − 0.996i)5-s − 0.413·6-s + 1.17i·7-s − 0.476i·8-s + 0.894·9-s + (−1.27 + 0.101i)10-s − 1.69·11-s + 0.203i·12-s + 0.346i·13-s + 1.49·14-s + (−0.323 + 0.0258i)15-s − 1.23·16-s + ⋯

Functional equation

Λ(s)=(1445s/2ΓC(s)L(s)=((0.07960.996i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1445s/2ΓC(s+1/2)L(s)=((0.07960.996i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14451445    =    51725 \cdot 17^{2}
Sign: 0.07960.996i-0.0796 - 0.996i
Analytic conductor: 11.538311.5383
Root analytic conductor: 3.396813.39681
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1445(579,)\chi_{1445} (579, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1445, ( :1/2), 0.07960.996i)(2,\ 1445,\ (\ :1/2),\ -0.0796 - 0.996i)

Particular Values

L(1)L(1) \approx 0.52819957470.5281995747
L(12)L(\frac12) \approx 0.52819957470.5281995747
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(0.178+2.22i)T 1 + (0.178 + 2.22i)T
17 1 1
good2 1+1.80iT2T2 1 + 1.80iT - 2T^{2}
3 1+0.561iT3T2 1 + 0.561iT - 3T^{2}
7 13.11iT7T2 1 - 3.11iT - 7T^{2}
11 1+5.61T+11T2 1 + 5.61T + 11T^{2}
13 11.24iT13T2 1 - 1.24iT - 13T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+2.32iT23T2 1 + 2.32iT - 23T^{2}
29 1+6.62T+29T2 1 + 6.62T + 29T^{2}
31 1+4.55T+31T2 1 + 4.55T + 31T^{2}
37 11.90iT37T2 1 - 1.90iT - 37T^{2}
41 1+5.92T+41T2 1 + 5.92T + 41T^{2}
43 12.04iT43T2 1 - 2.04iT - 43T^{2}
47 1+4.85iT47T2 1 + 4.85iT - 47T^{2}
53 1+9.11iT53T2 1 + 9.11iT - 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1+5.65T+61T2 1 + 5.65T + 61T^{2}
67 1+8.46iT67T2 1 + 8.46iT - 67T^{2}
71 18.79T+71T2 1 - 8.79T + 71T^{2}
73 11.56iT73T2 1 - 1.56iT - 73T^{2}
79 14.91T+79T2 1 - 4.91T + 79T^{2}
83 13.94iT83T2 1 - 3.94iT - 83T^{2}
89 110.6T+89T2 1 - 10.6T + 89T^{2}
97 1+12.3iT97T2 1 + 12.3iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.107423508847395826202542639438, −8.349077703114772634896844755732, −7.51509459494567180323475018110, −6.40102770068463814110379978601, −5.31506757686974333587922650625, −4.62450926391204792615116985542, −3.53701577490347268637141418718, −2.26278586586650891631245636533, −1.81163493190699557251385266472, −0.19059573793131521051834249299, 2.13380630298043453059573986944, 3.42702900355250888737398816116, 4.39174751880946961855759722451, 5.30969205085939295447547132646, 6.16924730688407643678359690870, 7.14496395030185312949922233893, 7.48101120322767627993840408366, 8.011886429356934704579080719973, 9.256694185231227719910744606296, 10.29127679603532216063845363353

Graph of the ZZ-function along the critical line