Properties

Degree $2$
Conductor $261$
Sign $0.187 + 0.982i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.488 − 0.776i)2-s + (1.37 − 2.84i)4-s + (1.98 + 0.453i)5-s + (9.56 − 4.60i)7-s + (−6.52 + 0.735i)8-s + (−0.616 − 1.76i)10-s + (11.6 + 1.31i)11-s + (−11.0 + 8.79i)13-s + (−8.24 − 5.18i)14-s + (−4.12 − 5.16i)16-s + (−0.154 + 0.154i)17-s + (20.3 − 7.12i)19-s + (4.00 − 5.02i)20-s + (−4.65 − 9.67i)22-s + (−1.51 − 6.62i)23-s + ⋯
L(s)  = 1  + (−0.244 − 0.388i)2-s + (0.342 − 0.711i)4-s + (0.396 + 0.0906i)5-s + (1.36 − 0.657i)7-s + (−0.815 + 0.0919i)8-s + (−0.0616 − 0.176i)10-s + (1.05 + 0.119i)11-s + (−0.848 + 0.676i)13-s + (−0.588 − 0.370i)14-s + (−0.257 − 0.323i)16-s + (−0.00910 + 0.00910i)17-s + (1.07 − 0.374i)19-s + (0.200 − 0.251i)20-s + (−0.211 − 0.439i)22-s + (−0.0657 − 0.288i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.187 + 0.982i$
Motivic weight: \(2\)
Character: $\chi_{261} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.40220 - 1.15933i\)
\(L(\frac12)\) \(\approx\) \(1.40220 - 1.15933i\)
\(L(2)\) 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 \( 1 + (-15.7 + 24.3i)T \)
good2 \( 1 + (0.488 + 0.776i)T + (-1.73 + 3.60i)T^{2} \)
5 \( 1 + (-1.98 - 0.453i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (-9.56 + 4.60i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (-11.6 - 1.31i)T + (117. + 26.9i)T^{2} \)
13 \( 1 + (11.0 - 8.79i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (0.154 - 0.154i)T - 289iT^{2} \)
19 \( 1 + (-20.3 + 7.12i)T + (282. - 225. i)T^{2} \)
23 \( 1 + (1.51 + 6.62i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (0.406 + 0.646i)T + (-416. + 865. i)T^{2} \)
37 \( 1 + (6.96 - 0.784i)T + (1.33e3 - 304. i)T^{2} \)
41 \( 1 + (35.8 + 35.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (-18.1 - 11.4i)T + (802. + 1.66e3i)T^{2} \)
47 \( 1 + (-2.57 + 22.8i)T + (-2.15e3 - 491. i)T^{2} \)
53 \( 1 + (12.7 - 55.8i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 - 48.4T + 3.48e3T^{2} \)
61 \( 1 + (24.4 - 69.8i)T + (-2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (-33.7 - 26.9i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (78.3 - 62.5i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (-5.59 + 8.90i)T + (-2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-4.65 - 41.2i)T + (-6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (85.5 + 41.1i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (-89.2 - 142. i)T + (-3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (-20.2 - 57.9i)T + (-7.35e3 + 5.86e3i)T^{2} \)
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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

−11.72735993967440247843353982696, −10.55354470430849728884508389455, −9.798130236269068273608074540384, −8.902399099497734565353752921387, −7.53702137984908068516321802508, −6.59887026883129365492495221297, −5.34192541614135185399767885811, −4.24819763488633976919574711508, −2.27168567884146388275167537217, −1.16125336938569648084395250928, 1.77675904286980953728425669193, 3.28354544778787083411275538539, 4.91113433767270089561834518099, 5.94285277343320149990309868406, 7.23011171092195063302940198853, 8.048827834695696990030649406079, 8.884089041906861973150569736963, 9.860008025792949864260627657955, 11.34450049501543737771614366633, 11.85101601468357456247477068197

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