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Properties

Label 2-1710-15.8-c1-0-9
Degree 22
Conductor 17101710
Sign −0.144−0.989i-0.144 - 0.989i
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.70 − 1.44i)5-s + (0.414 − 0.414i)7-s + (−0.707 + 0.707i)8-s + (−0.185 − 2.22i)10-s + 2.58i·11-s + (−2.88 − 2.88i)13-s + 0.585·14-s − 1.00·16-s + (5.60 + 5.60i)17-s + i·19-s + (1.44 − 1.70i)20-s + (−1.82 + 1.82i)22-s + (0.322 − 0.322i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.763 − 0.645i)5-s + (0.156 − 0.156i)7-s + (−0.250 + 0.250i)8-s + (−0.0587 − 0.704i)10-s + 0.779i·11-s + (−0.801 − 0.801i)13-s + 0.156·14-s − 0.250·16-s + (1.36 + 1.36i)17-s + 0.229i·19-s + (0.322 − 0.381i)20-s + (−0.389 + 0.389i)22-s + (0.0673 − 0.0673i)23-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=((−0.144−0.989i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+1/2) L(s)=((−0.144−0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: −0.144−0.989i-0.144 - 0.989i
Analytic conductor: 13.654413.6544
Root analytic conductor: 3.695183.69518
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1710(1673,⋅)\chi_{1710} (1673, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1710, ( :1/2), −0.144−0.989i)(2,\ 1710,\ (\ :1/2),\ -0.144 - 0.989i)

Particular Values

L(1)L(1) ≈\approx 1.6347286581.634728658
L(12)L(\frac12) ≈\approx 1.6347286581.634728658
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
3 1 1
5 1+(1.70+1.44i)T 1 + (1.70 + 1.44i)T
19 1−iT 1 - iT
good7 1+(−0.414+0.414i)T−7iT2 1 + (-0.414 + 0.414i)T - 7iT^{2}
11 1−2.58iT−11T2 1 - 2.58iT - 11T^{2}
13 1+(2.88+2.88i)T+13iT2 1 + (2.88 + 2.88i)T + 13iT^{2}
17 1+(−5.60−5.60i)T+17iT2 1 + (-5.60 - 5.60i)T + 17iT^{2}
23 1+(−0.322+0.322i)T−23iT2 1 + (-0.322 + 0.322i)T - 23iT^{2}
29 1−5.77T+29T2 1 - 5.77T + 29T^{2}
31 1+5.13T+31T2 1 + 5.13T + 31T^{2}
37 1+(1.41−1.41i)T−37iT2 1 + (1.41 - 1.41i)T - 37iT^{2}
41 1−5.47iT−41T2 1 - 5.47iT - 41T^{2}
43 1+(−2.47−2.47i)T+43iT2 1 + (-2.47 - 2.47i)T + 43iT^{2}
47 1+(−9.45−9.45i)T+47iT2 1 + (-9.45 - 9.45i)T + 47iT^{2}
53 1+(4.24−4.24i)T−53iT2 1 + (4.24 - 4.24i)T - 53iT^{2}
59 1−0.302T+59T2 1 - 0.302T + 59T^{2}
61 1−1.25T+61T2 1 - 1.25T + 61T^{2}
67 1+(−4.30+4.30i)T−67iT2 1 + (-4.30 + 4.30i)T - 67iT^{2}
71 1−7.25iT−71T2 1 - 7.25iT - 71T^{2}
73 1+(1.82+1.82i)T+73iT2 1 + (1.82 + 1.82i)T + 73iT^{2}
79 1−2.61iT−79T2 1 - 2.61iT - 79T^{2}
83 1+(−5.45+5.45i)T−83iT2 1 + (-5.45 + 5.45i)T - 83iT^{2}
89 1+7.43T+89T2 1 + 7.43T + 89T^{2}
97 1+(6.23−6.23i)T−97iT2 1 + (6.23 - 6.23i)T - 97iT^{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

−9.483772434400052406783223357095, −8.475875118925986862840490292652, −7.74650283204344074612155578966, −7.46160796852631665124684089226, −6.23274722833820964168609154032, −5.38564956094928921140459275261, −4.62716232800631871189312389932, −3.88546103137038238990265079400, −2.87270475536537822015484124224, −1.27451845574076676587608614906, 0.57382246165070478779006531283, 2.26587287078861357525870995790, 3.14724805119734103316840303134, 3.90975462724820163939800482527, 4.95989020606652820475778878729, 5.65760434549551029484279875490, 6.88247046386244857184599563256, 7.31228626366400702024643602929, 8.349088284322329198928725467763, 9.219505669826842761131119172751

Graph of the ZZ-function along the critical line