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Properties

Label 2-968-88.35-c1-0-1
Degree 22
Conductor 968968
Sign −0.909−0.416i-0.909 - 0.416i
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
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  + (−1.34 − 0.437i)2-s + (−0.0359 + 0.0261i)3-s + (1.61 + 1.17i)4-s + (0.0598 − 0.0194i)6-s + (−1.66 − 2.28i)8-s + (−0.926 + 2.85i)9-s − 0.0889·12-s + (1.23 + 3.80i)16-s + (−3.76 + 1.22i)17-s + (2.49 − 3.43i)18-s + (−2.65 − 3.66i)19-s + (0.119 + 0.0388i)24-s + (−4.04 + 2.93i)25-s + (−0.0824 − 0.253i)27-s − 5.65i·32-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.0207 + 0.0150i)3-s + (0.809 + 0.587i)4-s + (0.0244 − 0.00793i)6-s + (−0.587 − 0.809i)8-s + (−0.308 + 0.950i)9-s − 0.0256·12-s + (0.309 + 0.951i)16-s + (−0.913 + 0.296i)17-s + (0.587 − 0.808i)18-s + (−0.610 − 0.839i)19-s + (0.0244 + 0.00793i)24-s + (−0.809 + 0.587i)25-s + (−0.0158 − 0.0488i)27-s − 1.00i·32-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=((−0.909−0.416i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=((−0.909−0.416i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −0.909−0.416i-0.909 - 0.416i
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ968(475,⋅)\chi_{968} (475, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 968, ( :1/2), −0.909−0.416i)(2,\ 968,\ (\ :1/2),\ -0.909 - 0.416i)

Particular Values

L(1)L(1) ≈\approx 0.0381804+0.175068i0.0381804 + 0.175068i
L(12)L(\frac12) ≈\approx 0.0381804+0.175068i0.0381804 + 0.175068i
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+(1.34+0.437i)T 1 + (1.34 + 0.437i)T
11 1 1
good3 1+(0.0359−0.0261i)T+(0.927−2.85i)T2 1 + (0.0359 - 0.0261i)T + (0.927 - 2.85i)T^{2}
5 1+(4.04−2.93i)T2 1 + (4.04 - 2.93i)T^{2}
7 1+(2.16+6.65i)T2 1 + (2.16 + 6.65i)T^{2}
13 1+(−10.5−7.64i)T2 1 + (-10.5 - 7.64i)T^{2}
17 1+(3.76−1.22i)T+(13.7−9.99i)T2 1 + (3.76 - 1.22i)T + (13.7 - 9.99i)T^{2}
19 1+(2.65+3.66i)T+(−5.87+18.0i)T2 1 + (2.65 + 3.66i)T + (-5.87 + 18.0i)T^{2}
23 1−23T2 1 - 23T^{2}
29 1+(8.96+27.5i)T2 1 + (8.96 + 27.5i)T^{2}
31 1+(25.0+18.2i)T2 1 + (25.0 + 18.2i)T^{2}
37 1+(−11.4−35.1i)T2 1 + (-11.4 - 35.1i)T^{2}
41 1+(7.45+10.2i)T+(−12.6+38.9i)T2 1 + (7.45 + 10.2i)T + (-12.6 + 38.9i)T^{2}
43 1−12.7iT−43T2 1 - 12.7iT - 43T^{2}
47 1+(−14.5+44.6i)T2 1 + (-14.5 + 44.6i)T^{2}
53 1+(42.8+31.1i)T2 1 + (42.8 + 31.1i)T^{2}
59 1+(9.38+6.81i)T+(18.2+56.1i)T2 1 + (9.38 + 6.81i)T + (18.2 + 56.1i)T^{2}
61 1+(−49.3+35.8i)T2 1 + (-49.3 + 35.8i)T^{2}
67 1−12.3T+67T2 1 - 12.3T + 67T^{2}
71 1+(57.4−41.7i)T2 1 + (57.4 - 41.7i)T^{2}
73 1+(7.37−10.1i)T+(−22.5−69.4i)T2 1 + (7.37 - 10.1i)T + (-22.5 - 69.4i)T^{2}
79 1+(−63.9−46.4i)T2 1 + (-63.9 - 46.4i)T^{2}
83 1+(12.2−3.97i)T+(67.1−48.7i)T2 1 + (12.2 - 3.97i)T + (67.1 - 48.7i)T^{2}
89 1+17.8T+89T2 1 + 17.8T + 89T^{2}
97 1+(5.58−17.1i)T+(−78.4−57.0i)T2 1 + (5.58 - 17.1i)T + (-78.4 - 57.0i)T^{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

−10.37530492563879473086764010716, −9.544209524046389047759754334694, −8.675746660451813013180675088032, −8.095241886814141247249405871781, −7.16099410842319220092261860553, −6.36796155191010471947336625734, −5.19068869853605385350088027700, −3.99609730131615657956111421278, −2.70595147384849362482474060645, −1.78583644548192812089567404171, 0.10649088180158756138715827104, 1.71939691483490035977783903654, 2.99654650048010015027728634458, 4.31217694260404641171611152605, 5.68058520119245833170905205859, 6.37345876487170036472002987761, 7.12159486820145165426833879927, 8.183810376307093680571869611593, 8.768043636578451530712067860290, 9.604410166728345718024155946245

Graph of the ZZ-function along the critical line