Properties

Degree $2$
Conductor $261$
Sign $-0.126 + 0.992i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.42 − 2.27i)2-s + (−1.39 + 2.88i)4-s + (6.69 + 1.52i)5-s + (−4.78 + 2.30i)7-s + (−2.11 + 0.238i)8-s + (−6.09 − 17.4i)10-s + (3.10 + 0.349i)11-s + (14.9 − 11.9i)13-s + (12.0 + 7.57i)14-s + (11.5 + 14.5i)16-s + (19.1 − 19.1i)17-s + (3.26 − 1.14i)19-s + (−13.7 + 17.2i)20-s + (−3.63 − 7.54i)22-s + (3.11 + 13.6i)23-s + ⋯
L(s)  = 1  + (−0.714 − 1.13i)2-s + (−0.347 + 0.722i)4-s + (1.33 + 0.305i)5-s + (−0.682 + 0.328i)7-s + (−0.264 + 0.0298i)8-s + (−0.609 − 1.74i)10-s + (0.281 + 0.0317i)11-s + (1.14 − 0.915i)13-s + (0.861 + 0.541i)14-s + (0.722 + 0.906i)16-s + (1.12 − 1.12i)17-s + (0.171 − 0.0600i)19-s + (−0.686 + 0.861i)20-s + (−0.165 − 0.343i)22-s + (0.135 + 0.593i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.126 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.126 + 0.992i$
Motivic weight: \(2\)
Character: $\chi_{261} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.126 + 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.874395 - 0.992504i\)
\(L(\frac12)\) \(\approx\) \(0.874395 - 0.992504i\)
\(L(2)\) 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 \( 1 + (-22.5 + 18.2i)T \)
good2 \( 1 + (1.42 + 2.27i)T + (-1.73 + 3.60i)T^{2} \)
5 \( 1 + (-6.69 - 1.52i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (4.78 - 2.30i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (-3.10 - 0.349i)T + (117. + 26.9i)T^{2} \)
13 \( 1 + (-14.9 + 11.9i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (-19.1 + 19.1i)T - 289iT^{2} \)
19 \( 1 + (-3.26 + 1.14i)T + (282. - 225. i)T^{2} \)
23 \( 1 + (-3.11 - 13.6i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (30.9 + 49.2i)T + (-416. + 865. i)T^{2} \)
37 \( 1 + (-23.4 + 2.64i)T + (1.33e3 - 304. i)T^{2} \)
41 \( 1 + (29.0 + 29.0i)T + 1.68e3iT^{2} \)
43 \( 1 + (-28.3 - 17.8i)T + (802. + 1.66e3i)T^{2} \)
47 \( 1 + (5.42 - 48.1i)T + (-2.15e3 - 491. i)T^{2} \)
53 \( 1 + (2.93 - 12.8i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 + 11.7T + 3.48e3T^{2} \)
61 \( 1 + (18.2 - 52.2i)T + (-2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (6.83 + 5.44i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (-49.1 + 39.1i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (-33.0 + 52.6i)T + (-2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-5.05 - 44.8i)T + (-6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (-123. - 59.5i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (38.7 + 61.6i)T + (-3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (-16.9 - 48.4i)T + (-7.35e3 + 5.86e3i)T^{2} \)
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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

−11.27669710508990330010912454097, −10.43620227273340786290917825430, −9.549086773532373017006209317670, −9.297479430631713379101818663277, −7.85450228000694655023117275160, −6.21590627742536858276729512611, −5.63850442165175173005647980755, −3.39561606781404132926850044372, −2.47295067700204672185222127138, −1.02320390202698960026386308762, 1.39402196458551343439429538623, 3.48226392713498619558177403806, 5.36306106865431765871320016236, 6.31231788133476794100983295208, 6.79783619885718920490565880770, 8.278121049221364177629777215822, 9.011608328445517961512151732330, 9.792366358548425671101556923135, 10.62316715438273118969918524908, 12.19637533636820084251592850009

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