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Properties

Label 2-832-104.101-c1-0-18
Degree 22
Conductor 832832
Sign 0.869+0.494i0.869 + 0.494i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
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  + (1.16 − 0.669i)3-s + 3.88·5-s + (−2.20 − 1.27i)7-s + (−0.602 + 1.04i)9-s + (−0.571 − 0.990i)11-s + (1 + 3.46i)13-s + (4.50 − 2.60i)15-s + (3.44 − 5.96i)17-s + (3.93 − 6.81i)19-s − 3.40·21-s + (3.93 + 6.81i)23-s + 10.0·25-s + 5.63i·27-s + (−2.51 + 1.45i)29-s − 2i·31-s + ⋯
L(s)  = 1  + (0.669 − 0.386i)3-s + 1.73·5-s + (−0.832 − 0.480i)7-s + (−0.200 + 0.347i)9-s + (−0.172 − 0.298i)11-s + (0.277 + 0.960i)13-s + (1.16 − 0.671i)15-s + (0.834 − 1.44i)17-s + (0.902 − 1.56i)19-s − 0.744·21-s + (0.820 + 1.42i)23-s + 2.01·25-s + 1.08i·27-s + (−0.467 + 0.270i)29-s − 0.359i·31-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.869+0.494i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((0.869+0.494i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.869+0.494i0.869 + 0.494i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(673,⋅)\chi_{832} (673, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 0.869+0.494i)(2,\ 832,\ (\ :1/2),\ 0.869 + 0.494i)

Particular Values

L(1)L(1) ≈\approx 2.33496−0.617636i2.33496 - 0.617636i
L(12)L(\frac12) ≈\approx 2.33496−0.617636i2.33496 - 0.617636i
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
13 1+(−1−3.46i)T 1 + (-1 - 3.46i)T
good3 1+(−1.16+0.669i)T+(1.5−2.59i)T2 1 + (-1.16 + 0.669i)T + (1.5 - 2.59i)T^{2}
5 1−3.88T+5T2 1 - 3.88T + 5T^{2}
7 1+(2.20+1.27i)T+(3.5+6.06i)T2 1 + (2.20 + 1.27i)T + (3.5 + 6.06i)T^{2}
11 1+(0.571+0.990i)T+(−5.5+9.52i)T2 1 + (0.571 + 0.990i)T + (-5.5 + 9.52i)T^{2}
17 1+(−3.44+5.96i)T+(−8.5−14.7i)T2 1 + (-3.44 + 5.96i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.93+6.81i)T+(−9.5−16.4i)T2 1 + (-3.93 + 6.81i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.93−6.81i)T+(−11.5+19.9i)T2 1 + (-3.93 - 6.81i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.51−1.45i)T+(14.5−25.1i)T2 1 + (2.51 - 1.45i)T + (14.5 - 25.1i)T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1+(1.78+3.08i)T+(−18.5+32.0i)T2 1 + (1.78 + 3.08i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.32−0.765i)T+(20.5−35.5i)T2 1 + (1.32 - 0.765i)T + (20.5 - 35.5i)T^{2}
43 1+(7.88+4.55i)T+(21.5+37.2i)T2 1 + (7.88 + 4.55i)T + (21.5 + 37.2i)T^{2}
47 1+2.11iT−47T2 1 + 2.11iT - 47T^{2}
53 1−9.01iT−53T2 1 - 9.01iT - 53T^{2}
59 1+(2.79−4.83i)T+(−29.5−51.0i)T2 1 + (2.79 - 4.83i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.5+0.866i)T+(30.5+52.8i)T2 1 + (1.5 + 0.866i)T + (30.5 + 52.8i)T^{2}
67 1+(−2.20−3.81i)T+(−33.5+58.0i)T2 1 + (-2.20 - 3.81i)T + (-33.5 + 58.0i)T^{2}
71 1+(−3.24−1.87i)T+(35.5+61.4i)T2 1 + (-3.24 - 1.87i)T + (35.5 + 61.4i)T^{2}
73 1−11.3iT−73T2 1 - 11.3iT - 73T^{2}
79 1+7.90T+79T2 1 + 7.90T + 79T^{2}
83 1+8.10T+83T2 1 + 8.10T + 83T^{2}
89 1+(−2.51+1.45i)T+(44.5−77.0i)T2 1 + (-2.51 + 1.45i)T + (44.5 - 77.0i)T^{2}
97 1+(−1.5−0.866i)T+(48.5+84.0i)T2 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)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

−9.755159962108812489103450281697, −9.423442935699565624874176571896, −8.754802161430476856232624512143, −7.21630139756258986465983299841, −6.98855079199724988041186609847, −5.66293996122109900123101248699, −5.08229428260084848720907617240, −3.26921829795448337146472742727, −2.57110836353210612464246590619, −1.32546201315778917061428781216, 1.58485609718571830350414076191, 2.86837494947179096262979113107, 3.50605576327557643668248723002, 5.18163529754914898832567140484, 5.99858244959645947669211531268, 6.44199924181915001488702170303, 8.038014489630476176390560621871, 8.715766237538948984127176885873, 9.690287803272624712761754973841, 9.971537513020996681473802269725

Graph of the ZZ-function along the critical line