↑

Properties

Label 2-17-17.16-c15-0-14
Degree 22
Conductor 1717
Sign 0.858−0.512i0.858 - 0.512i
Analytic cond. 24.257824.2578
Root an. cond. 4.925234.92523
Motivic weight 1515
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 326.·2-s − 2.24e3i·3-s + 7.36e4·4-s + 2.24e5i·5-s − 7.30e5i·6-s + 1.90e6i·7-s + 1.33e7·8-s + 9.32e6·9-s + 7.33e7i·10-s + 3.89e7i·11-s − 1.65e8i·12-s − 1.61e8·13-s + 6.19e8i·14-s + 5.03e8·15-s + 1.93e9·16-s + (1.45e9 − 8.66e8i)17-s + ⋯
L(s)  = 1  + 1.80·2-s − 0.591i·3-s + 2.24·4-s + 1.28i·5-s − 1.06i·6-s + 0.872i·7-s + 2.24·8-s + 0.650·9-s + 2.31i·10-s + 0.602i·11-s − 1.32i·12-s − 0.713·13-s + 1.57i·14-s + 0.761·15-s + 1.80·16-s + (0.858 − 0.512i)17-s + ⋯

Functional equation

Λ(s)=(17s/2 ΓC(s) L(s)=((0.858−0.512i) Λ‾(16−s)\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(16-s) \end{aligned}
Λ(s)=(17s/2 ΓC(s+15/2) L(s)=((0.858−0.512i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1717
Sign: 0.858−0.512i0.858 - 0.512i
Analytic conductor: 24.257824.2578
Root analytic conductor: 4.925234.92523
Motivic weight: 1515
Rational: no
Arithmetic: yes
Character: χ17(16,⋅)\chi_{17} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 17, ( :15/2), 0.858−0.512i)(2,\ 17,\ (\ :15/2),\ 0.858 - 0.512i)

Particular Values

L(8)L(8) ≈\approx 6.0308577376.030857737
L(12)L(\frac12) ≈\approx 6.0308577376.030857737
L(172)L(\frac{17}{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)
bad17 1+(−1.45e9+8.66e8i)T 1 + (-1.45e9 + 8.66e8i)T
good2 1−326.T+3.27e4T2 1 - 326.T + 3.27e4T^{2}
3 1+2.24e3iT−1.43e7T2 1 + 2.24e3iT - 1.43e7T^{2}
5 1−2.24e5iT−3.05e10T2 1 - 2.24e5iT - 3.05e10T^{2}
7 1−1.90e6iT−4.74e12T2 1 - 1.90e6iT - 4.74e12T^{2}
11 1−3.89e7iT−4.17e15T2 1 - 3.89e7iT - 4.17e15T^{2}
13 1+1.61e8T+5.11e16T2 1 + 1.61e8T + 5.11e16T^{2}
19 1−1.70e9T+1.51e19T2 1 - 1.70e9T + 1.51e19T^{2}
23 1+2.85e10iT−2.66e20T2 1 + 2.85e10iT - 2.66e20T^{2}
29 1+2.44e10iT−8.62e21T2 1 + 2.44e10iT - 8.62e21T^{2}
31 1−2.75e11iT−2.34e22T2 1 - 2.75e11iT - 2.34e22T^{2}
37 1−5.05e10iT−3.33e23T2 1 - 5.05e10iT - 3.33e23T^{2}
41 1+1.19e12iT−1.55e24T2 1 + 1.19e12iT - 1.55e24T^{2}
43 1+9.60e11T+3.17e24T2 1 + 9.60e11T + 3.17e24T^{2}
47 1+1.33e12T+1.20e25T2 1 + 1.33e12T + 1.20e25T^{2}
53 1+1.42e13T+7.31e25T2 1 + 1.42e13T + 7.31e25T^{2}
59 1−3.43e13T+3.65e26T2 1 - 3.43e13T + 3.65e26T^{2}
61 1+3.02e13iT−6.02e26T2 1 + 3.02e13iT - 6.02e26T^{2}
67 1+8.15e13T+2.46e27T2 1 + 8.15e13T + 2.46e27T^{2}
71 1+1.45e13iT−5.87e27T2 1 + 1.45e13iT - 5.87e27T^{2}
73 1−7.56e12iT−8.90e27T2 1 - 7.56e12iT - 8.90e27T^{2}
79 1+2.54e14iT−2.91e28T2 1 + 2.54e14iT - 2.91e28T^{2}
83 1−1.50e14T+6.11e28T2 1 - 1.50e14T + 6.11e28T^{2}
89 1+3.18e14T+1.74e29T2 1 + 3.18e14T + 1.74e29T^{2}
97 1−9.45e14iT−6.33e29T2 1 - 9.45e14iT - 6.33e29T^{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

−14.84060902799546180003653737664, −14.15048214469853668095258639636, −12.61782530608958091773669371700, −11.94458308839524855739307058496, −10.34081688486697533497338124752, −7.30162850591679009965555427860, −6.46122436173923843237014929840, −4.90339260835715152284704461738, −3.10013032326691276885144255952, −2.10675694517766282358331810080, 1.30655157747277102605595633127, 3.55705611278455970896714645673, 4.54086785476175580412229710422, 5.58505355873626104733890548921, 7.51776648301407033685987023177, 9.837092501510907203783014984631, 11.49390152049023548652316132944, 12.78934385246879192296509552022, 13.58219417353255317573917155353, 14.96329486189237143862846159820

Graph of the ZZ-function along the critical line