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Properties

Label 2-1710-15.2-c1-0-19
Degree 22
Conductor 17101710
Sign 0.0618+0.998i0.0618 + 0.998i
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 + (−2.12 − 0.707i)5-s + (2 + 2i)7-s + (0.707 + 0.707i)8-s + (2 − 0.999i)10-s − 2.82i·11-s + (−3 + 3i)13-s − 2.82·14-s − 1.00·16-s − i·19-s + (−0.707 + 2.12i)20-s + (2.00 + 2.00i)22-s + (−5.65 − 5.65i)23-s + (3.99 + 3i)25-s − 4.24i·26-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.948 − 0.316i)5-s + (0.755 + 0.755i)7-s + (0.250 + 0.250i)8-s + (0.632 − 0.316i)10-s − 0.852i·11-s + (−0.832 + 0.832i)13-s − 0.755·14-s − 0.250·16-s − 0.229i·19-s + (−0.158 + 0.474i)20-s + (0.426 + 0.426i)22-s + (−1.17 − 1.17i)23-s + (0.799 + 0.600i)25-s − 0.832i·26-s + ⋯

Functional equation

Λ(s)=(1710s/2 ΓC(s) L(s)=((0.0618+0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1710s/2 ΓC(s+1/2) L(s)=((0.0618+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17101710    =    2⋅32⋅5⋅192 \cdot 3^{2} \cdot 5 \cdot 19
Sign: 0.0618+0.998i0.0618 + 0.998i
Analytic conductor: 13.654413.6544
Root analytic conductor: 3.695183.69518
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1710(647,⋅)\chi_{1710} (647, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1710, ( :1/2), 0.0618+0.998i)(2,\ 1710,\ (\ :1/2),\ 0.0618 + 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.54865015710.5486501571
L(12)L(\frac12) ≈\approx 0.54865015710.5486501571
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+(2.12+0.707i)T 1 + (2.12 + 0.707i)T
19 1+iT 1 + iT
good7 1+(−2−2i)T+7iT2 1 + (-2 - 2i)T + 7iT^{2}
11 1+2.82iT−11T2 1 + 2.82iT - 11T^{2}
13 1+(3−3i)T−13iT2 1 + (3 - 3i)T - 13iT^{2}
17 1−17iT2 1 - 17iT^{2}
23 1+(5.65+5.65i)T+23iT2 1 + (5.65 + 5.65i)T + 23iT^{2}
29 1−7.07T+29T2 1 - 7.07T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+(3+3i)T+37iT2 1 + (3 + 3i)T + 37iT^{2}
41 1+1.41iT−41T2 1 + 1.41iT - 41T^{2}
43 1+(6−6i)T−43iT2 1 + (6 - 6i)T - 43iT^{2}
47 1+(5.65−5.65i)T−47iT2 1 + (5.65 - 5.65i)T - 47iT^{2}
53 1+(5.65+5.65i)T+53iT2 1 + (5.65 + 5.65i)T + 53iT^{2}
59 1−5.65T+59T2 1 - 5.65T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1+(8+8i)T+67iT2 1 + (8 + 8i)T + 67iT^{2}
71 1+5.65iT−71T2 1 + 5.65iT - 71T^{2}
73 1+(−9+9i)T−73iT2 1 + (-9 + 9i)T - 73iT^{2}
79 1+16iT−79T2 1 + 16iT - 79T^{2}
83 1+83iT2 1 + 83iT^{2}
89 1+18.3T+89T2 1 + 18.3T + 89T^{2}
97 1+(−7−7i)T+97iT2 1 + (-7 - 7i)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

−8.839417723646397479043885564023, −8.322478198215168674013965784683, −7.84570857926723419564519456230, −6.78550936239116545886388740767, −6.07488761495743712451534246407, −4.88589990782095366600428767675, −4.49713307020630503630312822573, −3.05451318120014190496143404888, −1.81876391460743089314822041401, −0.26833607439626720332224403975, 1.22459535314453097559706804559, 2.51162161977308223877613774447, 3.58668152028716469566935709833, 4.39743288766483528255034956020, 5.17479640671696858050285887082, 6.68136747619906058427825735993, 7.41651333384420294403234307465, 7.954912004530892733170248703252, 8.506657923211663301721676259995, 9.957681796342704104628062666306

Graph of the ZZ-function along the critical line