Properties

Degree $2$
Conductor $261$
Sign $0.219 + 0.975i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.415 + 0.0467i)2-s + (−3.72 + 0.851i)4-s + (−0.738 − 0.589i)5-s + (−0.577 + 2.53i)7-s + (3.08 − 1.07i)8-s + (0.334 + 0.209i)10-s + (8.65 + 3.02i)11-s + (−4.51 − 9.37i)13-s + (0.121 − 1.07i)14-s + (12.5 − 6.04i)16-s + (−21.4 − 21.4i)17-s + (14.2 − 22.6i)19-s + (3.25 + 1.56i)20-s + (−3.73 − 0.851i)22-s + (2.27 + 2.85i)23-s + ⋯
L(s)  = 1  + (−0.207 + 0.0233i)2-s + (−0.932 + 0.212i)4-s + (−0.147 − 0.117i)5-s + (−0.0824 + 0.361i)7-s + (0.385 − 0.134i)8-s + (0.0334 + 0.0209i)10-s + (0.786 + 0.275i)11-s + (−0.347 − 0.720i)13-s + (0.00866 − 0.0769i)14-s + (0.784 − 0.377i)16-s + (−1.26 − 1.26i)17-s + (0.750 − 1.19i)19-s + (0.162 + 0.0784i)20-s + (−0.169 − 0.0387i)22-s + (0.0990 + 0.124i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.219 + 0.975i$
Motivic weight: \(2\)
Character: $\chi_{261} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ 0.219 + 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.661350 - 0.528918i\)
\(L(\frac12)\) \(\approx\) \(0.661350 - 0.528918i\)
\(L(2)\) 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 \( 1 + (-24.3 + 15.7i)T \)
good2 \( 1 + (0.415 - 0.0467i)T + (3.89 - 0.890i)T^{2} \)
5 \( 1 + (0.738 + 0.589i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (0.577 - 2.53i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (-8.65 - 3.02i)T + (94.6 + 75.4i)T^{2} \)
13 \( 1 + (4.51 + 9.37i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (21.4 + 21.4i)T + 289iT^{2} \)
19 \( 1 + (-14.2 + 22.6i)T + (-156. - 325. i)T^{2} \)
23 \( 1 + (-2.27 - 2.85i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (-21.7 + 2.44i)T + (936. - 213. i)T^{2} \)
37 \( 1 + (46.0 - 16.1i)T + (1.07e3 - 853. i)T^{2} \)
41 \( 1 + (-25.3 + 25.3i)T - 1.68e3iT^{2} \)
43 \( 1 + (-2.78 + 24.6i)T + (-1.80e3 - 411. i)T^{2} \)
47 \( 1 + (7.49 - 21.4i)T + (-1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-18.6 + 23.3i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 + 18.2T + 3.48e3T^{2} \)
61 \( 1 + (78.8 - 49.5i)T + (1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (-24.5 + 50.9i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (56.1 + 116. i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (-47.2 - 5.32i)T + (5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (13.7 + 39.2i)T + (-4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (-29.4 - 129. i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (32.3 - 3.64i)T + (7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (36.1 + 22.6i)T + (4.08e3 + 8.47e3i)T^{2} \)
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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

−11.75996412143553837419169762414, −10.45179292722445850626939202843, −9.368094864418911939729045190666, −8.887979386201773256334173011487, −7.73995025979421230703787960233, −6.67524905312953981975164200732, −5.14489653512444484310393685251, −4.33707106906199992035694991667, −2.78560209307350213371469119456, −0.52509818128629735051354513371, 1.44922845386352759999560596444, 3.66209079463125230559330132261, 4.52809630068803606973200545493, 5.91291054455584079210025819140, 7.03967863456899934988858134475, 8.301856813613751269433590679932, 9.058957640333961721786958539842, 9.996531084149996494186981579775, 10.87095781014542907853203579574, 11.96583039380450537143242319749

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