Properties

Degree $2$
Conductor $261$
Sign $0.998 + 0.0485i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.05 + 1.29i)2-s + (0.827 − 1.71i)4-s + (−5.87 − 1.34i)5-s + (−9.36 + 4.50i)7-s + (−0.569 − 5.05i)8-s + (13.8 − 4.83i)10-s + (−0.977 + 8.67i)11-s + (8.98 − 7.16i)13-s + (13.4 − 21.3i)14-s + (12.4 + 15.6i)16-s + (9.77 + 9.77i)17-s + (−4.49 − 12.8i)19-s + (−7.17 + 8.99i)20-s + (−9.20 − 19.1i)22-s + (−6.44 − 28.2i)23-s + ⋯
L(s)  = 1  + (−1.02 + 0.646i)2-s + (0.206 − 0.429i)4-s + (−1.17 − 0.268i)5-s + (−1.33 + 0.644i)7-s + (−0.0711 − 0.631i)8-s + (1.38 − 0.483i)10-s + (−0.0888 + 0.788i)11-s + (0.691 − 0.551i)13-s + (0.960 − 1.52i)14-s + (0.779 + 0.976i)16-s + (0.574 + 0.574i)17-s + (−0.236 − 0.675i)19-s + (−0.358 + 0.449i)20-s + (−0.418 − 0.868i)22-s + (−0.280 − 1.22i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0485i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.998 + 0.0485i$
Motivic weight: \(2\)
Character: $\chi_{261} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.998 + 0.0485i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.425037 - 0.0103138i\)
\(L(\frac12)\) \(\approx\) \(0.425037 - 0.0103138i\)
\(L(2)\) 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 \( 1 + (-28.9 + 1.48i)T \)
good2 \( 1 + (2.05 - 1.29i)T + (1.73 - 3.60i)T^{2} \)
5 \( 1 + (5.87 + 1.34i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (9.36 - 4.50i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (0.977 - 8.67i)T + (-117. - 26.9i)T^{2} \)
13 \( 1 + (-8.98 + 7.16i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (-9.77 - 9.77i)T + 289iT^{2} \)
19 \( 1 + (4.49 + 12.8i)T + (-282. + 225. i)T^{2} \)
23 \( 1 + (6.44 + 28.2i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (-32.8 + 20.6i)T + (416. - 865. i)T^{2} \)
37 \( 1 + (-6.61 - 58.7i)T + (-1.33e3 + 304. i)T^{2} \)
41 \( 1 + (28.8 - 28.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (-23.3 + 37.1i)T + (-802. - 1.66e3i)T^{2} \)
47 \( 1 + (-16.5 - 1.86i)T + (2.15e3 + 491. i)T^{2} \)
53 \( 1 + (-12.8 + 56.5i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 - 34.0T + 3.48e3T^{2} \)
61 \( 1 + (-5.90 - 2.06i)T + (2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (53.1 + 42.4i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (-24.4 + 19.5i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (43.4 + 27.2i)T + (2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (-36.9 + 4.16i)T + (6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (62.9 + 30.2i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (23.7 - 14.9i)T + (3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (33.9 - 11.8i)T + (7.35e3 - 5.86e3i)T^{2} \)
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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

−11.92635786497828421265921368099, −10.38670880160148029155027355319, −9.718448932864156595728570281544, −8.535892140797833484959427477058, −8.143800641116300609178598874143, −6.90982461963924133241647462737, −6.15621994908129591338343131633, −4.34997821551794953180968322656, −3.10955775134322709329130466755, −0.45422649201892534330915892658, 0.869585233720525472694418372977, 3.04760444681682925358154522716, 3.92836467999394615710158064653, 5.85235362304476427076190464826, 7.12449018866276938529093096193, 8.048032607223379126994280694453, 9.003207676774401706875243210620, 9.956054457924974398288676114152, 10.70669791725628719741733594215, 11.55774393277436064236540778894

Graph of the $Z$-function along the critical line