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Properties

Label 2-261-29.24-c1-0-2
Degree $2$
Conductor $261$
Sign $0.963 - 0.267i$
Analytic cond. $2.08409$
Root an. cond. $1.44363$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ 0.963 - 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44533 + 0.196562i\)
\(L(\frac12)\) \(\approx\) \(1.44533 + 0.196562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (-1.41 + 5.19i)T \)
good2 \( 1 + (0.110 + 0.138i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-2.15 - 2.69i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.844 - 3.70i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (3.84 + 1.85i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-4.18 - 2.01i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 3.07T + 17T^{2} \)
19 \( 1 + (-0.799 + 3.50i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-0.270 + 0.339i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (2.81 + 3.53i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (5.70 - 2.74i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 - 1.97T + 41T^{2} \)
43 \( 1 + (0.156 - 0.196i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (4.33 + 2.08i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (6.83 + 8.57i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 6.06T + 59T^{2} \)
61 \( 1 + (0.843 + 3.69i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-4.74 + 2.28i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (5.25 + 2.53i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (6.57 - 8.24i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (9.04 - 4.35i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (1.57 - 6.92i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-8.71 - 10.9i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-3.18 + 13.9i)T + (-87.3 - 42.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line