Properties

Degree $2$
Conductor $261$
Sign $-0.0648 + 0.997i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.63 − 2.59i)2-s + (−2.35 − 4.88i)4-s + (4.43 − 1.01i)5-s + (10.7 + 5.18i)7-s + (−4.32 − 0.487i)8-s + (4.61 − 13.1i)10-s + (−2.90 + 0.326i)11-s + (2.24 + 1.79i)13-s + (31.0 − 19.5i)14-s + (5.18 − 6.49i)16-s + (−2.44 − 2.44i)17-s + (−31.9 − 11.1i)19-s + (−15.3 − 19.2i)20-s + (−3.88 + 8.07i)22-s + (−9.24 + 40.5i)23-s + ⋯
L(s)  = 1  + (0.816 − 1.29i)2-s + (−0.587 − 1.22i)4-s + (0.887 − 0.202i)5-s + (1.53 + 0.740i)7-s + (−0.540 − 0.0609i)8-s + (0.461 − 1.31i)10-s + (−0.263 + 0.0297i)11-s + (0.173 + 0.137i)13-s + (2.21 − 1.39i)14-s + (0.323 − 0.406i)16-s + (−0.144 − 0.144i)17-s + (−1.68 − 0.588i)19-s + (−0.769 − 0.964i)20-s + (−0.176 + 0.366i)22-s + (−0.402 + 1.76i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0648 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0648 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.0648 + 0.997i$
Motivic weight: \(2\)
Character: $\chi_{261} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.0648 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.08485 - 2.22470i\)
\(L(\frac12)\) \(\approx\) \(2.08485 - 2.22470i\)
\(L(2)\) 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 \( 1 + (18.6 + 22.1i)T \)
good2 \( 1 + (-1.63 + 2.59i)T + (-1.73 - 3.60i)T^{2} \)
5 \( 1 + (-4.43 + 1.01i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + (-10.7 - 5.18i)T + (30.5 + 38.3i)T^{2} \)
11 \( 1 + (2.90 - 0.326i)T + (117. - 26.9i)T^{2} \)
13 \( 1 + (-2.24 - 1.79i)T + (37.6 + 164. i)T^{2} \)
17 \( 1 + (2.44 + 2.44i)T + 289iT^{2} \)
19 \( 1 + (31.9 + 11.1i)T + (282. + 225. i)T^{2} \)
23 \( 1 + (9.24 - 40.5i)T + (-476. - 229. i)T^{2} \)
31 \( 1 + (-7.40 + 11.7i)T + (-416. - 865. i)T^{2} \)
37 \( 1 + (19.6 + 2.21i)T + (1.33e3 + 304. i)T^{2} \)
41 \( 1 + (-15.2 + 15.2i)T - 1.68e3iT^{2} \)
43 \( 1 + (-1.58 + 0.996i)T + (802. - 1.66e3i)T^{2} \)
47 \( 1 + (-0.858 - 7.61i)T + (-2.15e3 + 491. i)T^{2} \)
53 \( 1 + (-7.13 - 31.2i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 - 9.47T + 3.48e3T^{2} \)
61 \( 1 + (19.5 + 55.9i)T + (-2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (45.3 - 36.1i)T + (998. - 4.37e3i)T^{2} \)
71 \( 1 + (-61.2 - 48.8i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (-40.7 - 64.8i)T + (-2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (-10.0 + 89.0i)T + (-6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (42.5 - 20.4i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (-31.5 + 50.2i)T + (-3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (41.8 - 119. i)T + (-7.35e3 - 5.86e3i)T^{2} \)
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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

−11.44891985497731571674863398652, −10.98270872834117906926221180035, −9.859758270672257480137156991760, −8.892432063055210314104403637202, −7.73467546632047408520864768238, −5.86977462816311988146116621688, −5.11660867989969669809664393246, −4.09918749164545477369985064436, −2.34190406427340968614934777042, −1.69555688287061144396905382783, 1.90982940106309203151152164798, 4.10854235768937376273528453812, 4.91930973259582578598642656793, 5.98578392607267570925834189314, 6.82055650371444344127488032077, 7.965285579205672355139193117440, 8.575507162585676480465399521931, 10.38521345954265518586833897080, 10.82113401210467755592240515042, 12.39595033603462685825841451042

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