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Properties

Label 2-1710-5.4-c1-0-1
Degree 22
Conductor 17101710
Sign −0.970−0.241i-0.970 - 0.241i
Analytic cond. 13.654413.6544
Root an. cond. 3.695183.69518
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  + i·2-s − 4-s + (0.539 − 2.17i)5-s − 1.07i·7-s − i·8-s + (2.17 + 0.539i)10-s − 6.34·11-s − 3.41i·13-s + 1.07·14-s + 16-s + 5.41i·17-s − 19-s + (−0.539 + 2.17i)20-s − 6.34i·22-s + 6.34i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.241 − 0.970i)5-s − 0.407i·7-s − 0.353i·8-s + (0.686 + 0.170i)10-s − 1.91·11-s − 0.948i·13-s + 0.288·14-s + 0.250·16-s + 1.31i·17-s − 0.229·19-s + (−0.120 + 0.485i)20-s − 1.35i·22-s + 1.32i·23-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=((−0.970−0.241i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+1/2) L(s)=((−0.970−0.241i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: −0.970−0.241i-0.970 - 0.241i
Analytic conductor: 13.654413.6544
Root analytic conductor: 3.695183.69518
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1710(1369,⋅)\chi_{1710} (1369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1710, ( :1/2), −0.970−0.241i)(2,\ 1710,\ (\ :1/2),\ -0.970 - 0.241i)

Particular Values

L(1)L(1) ≈\approx 0.32948505070.3294850507
L(12)L(\frac12) ≈\approx 0.32948505070.3294850507
L(32)L(\frac{3}{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−iT 1 - iT
3 1 1
5 1+(−0.539+2.17i)T 1 + (-0.539 + 2.17i)T
19 1+T 1 + T
good7 1+1.07iT−7T2 1 + 1.07iT - 7T^{2}
11 1+6.34T+11T2 1 + 6.34T + 11T^{2}
13 1+3.41iT−13T2 1 + 3.41iT - 13T^{2}
17 1−5.41iT−17T2 1 - 5.41iT - 17T^{2}
23 1−6.34iT−23T2 1 - 6.34iT - 23T^{2}
29 1+0.340T+29T2 1 + 0.340T + 29T^{2}
31 1−1.07T+31T2 1 - 1.07T + 31T^{2}
37 1−3.41iT−37T2 1 - 3.41iT - 37T^{2}
41 1+7.60T+41T2 1 + 7.60T + 41T^{2}
43 1−11.1iT−43T2 1 - 11.1iT - 43T^{2}
47 1−6.34iT−47T2 1 - 6.34iT - 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1−0.738T+59T2 1 - 0.738T + 59T^{2}
61 1+2.68T+61T2 1 + 2.68T + 61T^{2}
67 1−2.83iT−67T2 1 - 2.83iT - 67T^{2}
71 1−2.83T+71T2 1 - 2.83T + 71T^{2}
73 1+6.83iT−73T2 1 + 6.83iT - 73T^{2}
79 1−1.07T+79T2 1 - 1.07T + 79T^{2}
83 1−0.894iT−83T2 1 - 0.894iT - 83T^{2}
89 1−6.92T+89T2 1 - 6.92T + 89T^{2}
97 1+3.65iT−97T2 1 + 3.65iT - 97T^{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

−9.677032529712557563673202580562, −8.670691663987296803452966326415, −7.85454469725372450049387134052, −7.76909705346316116099607958637, −6.34437325228248525893757697690, −5.54539252209782841888999062270, −5.05096850719533793167359295554, −4.07471786719990899263776790590, −2.89080992944744505144796786435, −1.38079421182201414411848402703, 0.12179180070479452303132482089, 2.26031643762083475158380131555, 2.54594996841195646948524675541, 3.66911177223457823826096905366, 4.88668386644369064893215520033, 5.50612968597535100727602125238, 6.65914886377827287956507054772, 7.34270975495614874318811892179, 8.310827957019065320787657143628, 9.075648640791609283388760184347

Graph of the ZZ-function along the critical line