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Properties

Label 2-261-29.24-c1-0-2
Degree 22
Conductor 261261
Sign 0.963−0.267i0.963 - 0.267i
Analytic cond. 2.084092.08409
Root an. cond. 1.443631.44363
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  + (−0.110 − 0.138i)2-s + (0.438 − 1.91i)4-s + (2.15 + 2.69i)5-s + (0.844 + 3.70i)7-s + (−0.633 + 0.304i)8-s + (0.135 − 0.595i)10-s + (−3.84 − 1.85i)11-s + (4.18 + 2.01i)13-s + (0.419 − 0.525i)14-s + (−3.43 − 1.65i)16-s + 3.07·17-s + (0.799 − 3.50i)19-s + (6.12 − 2.94i)20-s + (0.168 + 0.736i)22-s + (0.270 − 0.339i)23-s + ⋯
L(s)  = 1  + (−0.0780 − 0.0978i)2-s + (0.219 − 0.959i)4-s + (0.962 + 1.20i)5-s + (0.319 + 1.39i)7-s + (−0.223 + 0.107i)8-s + (0.0430 − 0.188i)10-s + (−1.15 − 0.558i)11-s + (1.16 + 0.558i)13-s + (0.111 − 0.140i)14-s + (−0.858 − 0.413i)16-s + 0.745·17-s + (0.183 − 0.803i)19-s + (1.36 − 0.659i)20-s + (0.0358 + 0.157i)22-s + (0.0563 − 0.0706i)23-s + ⋯

Functional equation

Λ(s)=(261s/2 ΓC(s) L(s)=((0.963−0.267i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(261s/2 ΓC(s+1/2) L(s)=((0.963−0.267i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 261261    =    32⋅293^{2} \cdot 29
Sign: 0.963−0.267i0.963 - 0.267i
Analytic conductor: 2.084092.08409
Root analytic conductor: 1.443631.44363
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ261(82,⋅)\chi_{261} (82, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 261, ( :1/2), 0.963−0.267i)(2,\ 261,\ (\ :1/2),\ 0.963 - 0.267i)

Particular Values

L(1)L(1) ≈\approx 1.44533+0.196562i1.44533 + 0.196562i
L(12)L(\frac12) ≈\approx 1.44533+0.196562i1.44533 + 0.196562i
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
29 1+(−1.41+5.19i)T 1 + (-1.41 + 5.19i)T
good2 1+(0.110+0.138i)T+(−0.445+1.94i)T2 1 + (0.110 + 0.138i)T + (-0.445 + 1.94i)T^{2}
5 1+(−2.15−2.69i)T+(−1.11+4.87i)T2 1 + (-2.15 - 2.69i)T + (-1.11 + 4.87i)T^{2}
7 1+(−0.844−3.70i)T+(−6.30+3.03i)T2 1 + (-0.844 - 3.70i)T + (-6.30 + 3.03i)T^{2}
11 1+(3.84+1.85i)T+(6.85+8.60i)T2 1 + (3.84 + 1.85i)T + (6.85 + 8.60i)T^{2}
13 1+(−4.18−2.01i)T+(8.10+10.1i)T2 1 + (-4.18 - 2.01i)T + (8.10 + 10.1i)T^{2}
17 1−3.07T+17T2 1 - 3.07T + 17T^{2}
19 1+(−0.799+3.50i)T+(−17.1−8.24i)T2 1 + (-0.799 + 3.50i)T + (-17.1 - 8.24i)T^{2}
23 1+(−0.270+0.339i)T+(−5.11−22.4i)T2 1 + (-0.270 + 0.339i)T + (-5.11 - 22.4i)T^{2}
31 1+(2.81+3.53i)T+(−6.89+30.2i)T2 1 + (2.81 + 3.53i)T + (-6.89 + 30.2i)T^{2}
37 1+(5.70−2.74i)T+(23.0−28.9i)T2 1 + (5.70 - 2.74i)T + (23.0 - 28.9i)T^{2}
41 1−1.97T+41T2 1 - 1.97T + 41T^{2}
43 1+(0.156−0.196i)T+(−9.56−41.9i)T2 1 + (0.156 - 0.196i)T + (-9.56 - 41.9i)T^{2}
47 1+(4.33+2.08i)T+(29.3+36.7i)T2 1 + (4.33 + 2.08i)T + (29.3 + 36.7i)T^{2}
53 1+(6.83+8.57i)T+(−11.7+51.6i)T2 1 + (6.83 + 8.57i)T + (-11.7 + 51.6i)T^{2}
59 1−6.06T+59T2 1 - 6.06T + 59T^{2}
61 1+(0.843+3.69i)T+(−54.9+26.4i)T2 1 + (0.843 + 3.69i)T + (-54.9 + 26.4i)T^{2}
67 1+(−4.74+2.28i)T+(41.7−52.3i)T2 1 + (-4.74 + 2.28i)T + (41.7 - 52.3i)T^{2}
71 1+(5.25+2.53i)T+(44.2+55.5i)T2 1 + (5.25 + 2.53i)T + (44.2 + 55.5i)T^{2}
73 1+(6.57−8.24i)T+(−16.2−71.1i)T2 1 + (6.57 - 8.24i)T + (-16.2 - 71.1i)T^{2}
79 1+(9.04−4.35i)T+(49.2−61.7i)T2 1 + (9.04 - 4.35i)T + (49.2 - 61.7i)T^{2}
83 1+(1.57−6.92i)T+(−74.7−36.0i)T2 1 + (1.57 - 6.92i)T + (-74.7 - 36.0i)T^{2}
89 1+(−8.71−10.9i)T+(−19.8+86.7i)T2 1 + (-8.71 - 10.9i)T + (-19.8 + 86.7i)T^{2}
97 1+(−3.18+13.9i)T+(−87.3−42.0i)T2 1 + (-3.18 + 13.9i)T + (-87.3 - 42.0i)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

−11.60132910668065724127296640186, −11.06026335655543488123872417564, −10.18111139061576900955642575523, −9.362813434870077247038287621415, −8.294523923876114381922006250238, −6.71954446070904442155933916274, −5.89398536865674917273778821446, −5.30426225005972951697283723096, −2.93176896893100901980777196913, −2.00158413979407317273950138534, 1.45575751365569189982301071182, 3.40300949412256763896991595453, 4.68295842699002838279514513147, 5.76734204655947021558755475437, 7.26011920485863402095577490871, 8.023168323911245057196558657666, 8.884429650578294598686819185143, 10.14512838364414918480886754832, 10.79391878563521841195281360844, 12.24202746111963339147350100972

Graph of the ZZ-function along the critical line