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Properties

Label 2-3e4-81.25-c1-0-3
Degree 22
Conductor 8181
Sign 0.913−0.405i0.913 - 0.405i
Analytic cond. 0.6467880.646788
Root an. cond. 0.8042310.804231
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  + (−2.38 + 1.19i)2-s + (1.71 + 0.238i)3-s + (3.06 − 4.11i)4-s + (0.727 − 2.42i)5-s + (−4.37 + 1.48i)6-s + (0.202 − 0.469i)7-s + (−1.44 + 8.22i)8-s + (2.88 + 0.817i)9-s + (1.17 + 6.66i)10-s + (1.49 + 0.354i)11-s + (6.23 − 6.32i)12-s + (−4.71 + 3.09i)13-s + (0.0793 + 1.36i)14-s + (1.82 − 3.99i)15-s + (−3.45 − 11.5i)16-s + (−1.33 − 0.484i)17-s + ⋯
L(s)  = 1  + (−1.68 + 0.847i)2-s + (0.990 + 0.137i)3-s + (1.53 − 2.05i)4-s + (0.325 − 1.08i)5-s + (−1.78 + 0.607i)6-s + (0.0764 − 0.177i)7-s + (−0.512 + 2.90i)8-s + (0.962 + 0.272i)9-s + (0.371 + 2.10i)10-s + (0.450 + 0.106i)11-s + (1.79 − 1.82i)12-s + (−1.30 + 0.859i)13-s + (0.0211 + 0.363i)14-s + (0.471 − 1.03i)15-s + (−0.862 − 2.88i)16-s + (−0.322 − 0.117i)17-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((0.913−0.405i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+1/2) L(s)=((0.913−0.405i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: 0.913−0.405i0.913 - 0.405i
Analytic conductor: 0.6467880.646788
Root analytic conductor: 0.8042310.804231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ81(25,⋅)\chi_{81} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 81, ( :1/2), 0.913−0.405i)(2,\ 81,\ (\ :1/2),\ 0.913 - 0.405i)

Particular Values

L(1)L(1) ≈\approx 0.635427+0.134736i0.635427 + 0.134736i
L(12)L(\frac12) ≈\approx 0.635427+0.134736i0.635427 + 0.134736i
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)
bad3 1+(−1.71−0.238i)T 1 + (-1.71 - 0.238i)T
good2 1+(2.38−1.19i)T+(1.19−1.60i)T2 1 + (2.38 - 1.19i)T + (1.19 - 1.60i)T^{2}
5 1+(−0.727+2.42i)T+(−4.17−2.74i)T2 1 + (-0.727 + 2.42i)T + (-4.17 - 2.74i)T^{2}
7 1+(−0.202+0.469i)T+(−4.80−5.09i)T2 1 + (-0.202 + 0.469i)T + (-4.80 - 5.09i)T^{2}
11 1+(−1.49−0.354i)T+(9.82+4.93i)T2 1 + (-1.49 - 0.354i)T + (9.82 + 4.93i)T^{2}
13 1+(4.71−3.09i)T+(5.14−11.9i)T2 1 + (4.71 - 3.09i)T + (5.14 - 11.9i)T^{2}
17 1+(1.33+0.484i)T+(13.0+10.9i)T2 1 + (1.33 + 0.484i)T + (13.0 + 10.9i)T^{2}
19 1+(−0.986+0.359i)T+(14.5−12.2i)T2 1 + (-0.986 + 0.359i)T + (14.5 - 12.2i)T^{2}
23 1+(−0.103−0.239i)T+(−15.7+16.7i)T2 1 + (-0.103 - 0.239i)T + (-15.7 + 16.7i)T^{2}
29 1+(0.103−1.77i)T+(−28.8−3.36i)T2 1 + (0.103 - 1.77i)T + (-28.8 - 3.36i)T^{2}
31 1+(5.13+0.599i)T+(30.1+7.14i)T2 1 + (5.13 + 0.599i)T + (30.1 + 7.14i)T^{2}
37 1+(5.04−4.23i)T+(6.42−36.4i)T2 1 + (5.04 - 4.23i)T + (6.42 - 36.4i)T^{2}
41 1+(6.03+3.03i)T+(24.4+32.8i)T2 1 + (6.03 + 3.03i)T + (24.4 + 32.8i)T^{2}
43 1+(−4.19+4.44i)T+(−2.50−42.9i)T2 1 + (-4.19 + 4.44i)T + (-2.50 - 42.9i)T^{2}
47 1+(7.26−0.848i)T+(45.7−10.8i)T2 1 + (7.26 - 0.848i)T + (45.7 - 10.8i)T^{2}
53 1+(−4.74−8.21i)T+(−26.5+45.8i)T2 1 + (-4.74 - 8.21i)T + (-26.5 + 45.8i)T^{2}
59 1+(4.82−1.14i)T+(52.7−26.4i)T2 1 + (4.82 - 1.14i)T + (52.7 - 26.4i)T^{2}
61 1+(7.70+10.3i)T+(−17.4+58.4i)T2 1 + (7.70 + 10.3i)T + (-17.4 + 58.4i)T^{2}
67 1+(0.373+6.40i)T+(−66.5+7.77i)T2 1 + (0.373 + 6.40i)T + (-66.5 + 7.77i)T^{2}
71 1+(0.896+5.08i)T+(−66.7+24.2i)T2 1 + (0.896 + 5.08i)T + (-66.7 + 24.2i)T^{2}
73 1+(1.03−5.87i)T+(−68.5−24.9i)T2 1 + (1.03 - 5.87i)T + (-68.5 - 24.9i)T^{2}
79 1+(−8.88+4.46i)T+(47.1−63.3i)T2 1 + (-8.88 + 4.46i)T + (47.1 - 63.3i)T^{2}
83 1+(−11.5+5.81i)T+(49.5−66.5i)T2 1 + (-11.5 + 5.81i)T + (49.5 - 66.5i)T^{2}
89 1+(1.71−9.71i)T+(−83.6−30.4i)T2 1 + (1.71 - 9.71i)T + (-83.6 - 30.4i)T^{2}
97 1+(−1.03−3.44i)T+(−81.0+53.3i)T2 1 + (-1.03 - 3.44i)T + (-81.0 + 53.3i)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

−14.75741118556724189615965075556, −13.80364620890676650629480692807, −12.17489699228281643123589045754, −10.49517277142775370025351514899, −9.213650461005199371679324396116, −9.158984850424488719297315832303, −7.80711125731332191315600769670, −6.84795445479317731198182949351, −4.94988147862899352730689825929, −1.78469531244498435919842016086, 2.19511514743989928642532476910, 3.27278229191209588631189980987, 6.87117131731274078451387433775, 7.71204132256759362126205523326, 8.873487758739507447784824122875, 9.875789373439421969078153618594, 10.54416872078330315062874548976, 11.83453856866474517104425680103, 12.93474703720603036537414548147, 14.45824782597288216743762711666

Graph of the ZZ-function along the critical line