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Properties

Label 2-832-52.47-c1-0-9
Degree 22
Conductor 832832
Sign 0.176−0.984i0.176 - 0.984i
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  + i·3-s + (1.58 + 1.58i)5-s + (1.58 + 1.58i)7-s + 2·9-s + (4.16 + 4.16i)11-s + (−3.58 + 0.418i)13-s + (−1.58 + 1.58i)15-s − 7.32i·17-s + (1.16 − 1.16i)19-s + (−1.58 + 1.58i)21-s − 7.16·23-s + 5i·27-s + 1.16·29-s + (1.16 − 1.16i)31-s + (−4.16 + 4.16i)33-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.707 + 0.707i)5-s + (0.597 + 0.597i)7-s + 0.666·9-s + (1.25 + 1.25i)11-s + (−0.993 + 0.116i)13-s + (−0.408 + 0.408i)15-s − 1.77i·17-s + (0.266 − 0.266i)19-s + (−0.345 + 0.345i)21-s − 1.49·23-s + 0.962i·27-s + 0.215·29-s + (0.208 − 0.208i)31-s + (−0.724 + 0.724i)33-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.176−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((0.176−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.176−0.984i0.176 - 0.984i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(255,⋅)\chi_{832} (255, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 0.176−0.984i)(2,\ 832,\ (\ :1/2),\ 0.176 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.53429+1.28346i1.53429 + 1.28346i
L(12)L(\frac12) ≈\approx 1.53429+1.28346i1.53429 + 1.28346i
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+(3.58−0.418i)T 1 + (3.58 - 0.418i)T
good3 1−iT−3T2 1 - iT - 3T^{2}
5 1+(−1.58−1.58i)T+5iT2 1 + (-1.58 - 1.58i)T + 5iT^{2}
7 1+(−1.58−1.58i)T+7iT2 1 + (-1.58 - 1.58i)T + 7iT^{2}
11 1+(−4.16−4.16i)T+11iT2 1 + (-4.16 - 4.16i)T + 11iT^{2}
17 1+7.32iT−17T2 1 + 7.32iT - 17T^{2}
19 1+(−1.16+1.16i)T−19iT2 1 + (-1.16 + 1.16i)T - 19iT^{2}
23 1+7.16T+23T2 1 + 7.16T + 23T^{2}
29 1−1.16T+29T2 1 - 1.16T + 29T^{2}
31 1+(−1.16+1.16i)T−31iT2 1 + (-1.16 + 1.16i)T - 31iT^{2}
37 1+(−3.58+3.58i)T−37iT2 1 + (-3.58 + 3.58i)T - 37iT^{2}
41 1+(−5.16−5.16i)T+41iT2 1 + (-5.16 - 5.16i)T + 41iT^{2}
43 1+5T+43T2 1 + 5T + 43T^{2}
47 1+(−6.74−6.74i)T+47iT2 1 + (-6.74 - 6.74i)T + 47iT^{2}
53 1+9.48T+53T2 1 + 9.48T + 53T^{2}
59 1+(4+4i)T+59iT2 1 + (4 + 4i)T + 59iT^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+(7.32−7.32i)T−67iT2 1 + (7.32 - 7.32i)T - 67iT^{2}
71 1+(−1.58+1.58i)T−71iT2 1 + (-1.58 + 1.58i)T - 71iT^{2}
73 1+(6−6i)T−73iT2 1 + (6 - 6i)T - 73iT^{2}
79 1+3.48iT−79T2 1 + 3.48iT - 79T^{2}
83 1+(−5.83+5.83i)T−83iT2 1 + (-5.83 + 5.83i)T - 83iT^{2}
89 1+(2.83−2.83i)T−89iT2 1 + (2.83 - 2.83i)T - 89iT^{2}
97 1+(3.83+3.83i)T+97iT2 1 + (3.83 + 3.83i)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

−10.00628198435242968847048871249, −9.730510680209070965464954466179, −9.124698414770746559910754034372, −7.62717068919599713478781513719, −7.01570927895490141020769344955, −6.06415238159019724414418488628, −4.83577766874624744071703040557, −4.33049498478028715275291539062, −2.73850132333793788318946288493, −1.80589258387814198434170554441, 1.13055972747717683391360899055, 1.87662111953807492038533167085, 3.72861615547987754397050507290, 4.56478013553522972578217505363, 5.83980890013127581453489487632, 6.40678641852645980965417642592, 7.58676259482904407910120601333, 8.247893994105642261646282968000, 9.151282269567431503194652708632, 10.03612511894147779435084905340

Graph of the ZZ-function along the critical line