Properties

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

Origins

Related objects

Downloads

Learn more about

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: $\chi_{261} (118, \cdot )$
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 \( 1 + (36.1 - 22.6i)T + (4.08e3 - 8.47e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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