Properties

Degree $2$
Conductor $261$
Sign $-0.0769 + 0.997i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (3.24 − 2.03i)2-s + (4.63 − 9.61i)4-s + (5.12 + 1.17i)5-s + (−6.56 + 3.16i)7-s + (−2.86 − 25.4i)8-s + (19.0 − 6.65i)10-s + (0.480 − 4.26i)11-s + (2.73 − 2.17i)13-s + (−14.8 + 23.6i)14-s + (−34.4 − 43.2i)16-s + (6.15 + 6.15i)17-s + (5.18 + 14.8i)19-s + (35.0 − 43.9i)20-s + (−7.13 − 14.8i)22-s + (3.58 + 15.7i)23-s + ⋯
L(s)  = 1  + (1.62 − 1.01i)2-s + (1.15 − 2.40i)4-s + (1.02 + 0.234i)5-s + (−0.938 + 0.451i)7-s + (−0.357 − 3.17i)8-s + (1.90 − 0.665i)10-s + (0.0437 − 0.387i)11-s + (0.210 − 0.167i)13-s + (−1.06 + 1.68i)14-s + (−2.15 − 2.70i)16-s + (0.362 + 0.362i)17-s + (0.272 + 0.779i)19-s + (1.75 − 2.19i)20-s + (−0.324 − 0.673i)22-s + (0.156 + 0.683i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0769 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0769 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.0769 + 0.997i$
Motivic weight: \(2\)
Character: $\chi_{261} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.0769 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.78487 - 3.00821i\)
\(L(\frac12)\) \(\approx\) \(2.78487 - 3.00821i\)
\(L(2)\) 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 \( 1 + (-0.661 - 28.9i)T \)
good2 \( 1 + (-3.24 + 2.03i)T + (1.73 - 3.60i)T^{2} \)
5 \( 1 + (-5.12 - 1.17i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (6.56 - 3.16i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (-0.480 + 4.26i)T + (-117. - 26.9i)T^{2} \)
13 \( 1 + (-2.73 + 2.17i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (-6.15 - 6.15i)T + 289iT^{2} \)
19 \( 1 + (-5.18 - 14.8i)T + (-282. + 225. i)T^{2} \)
23 \( 1 + (-3.58 - 15.7i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (34.6 - 21.8i)T + (416. - 865. i)T^{2} \)
37 \( 1 + (5.62 + 49.9i)T + (-1.33e3 + 304. i)T^{2} \)
41 \( 1 + (-0.940 + 0.940i)T - 1.68e3iT^{2} \)
43 \( 1 + (10.2 - 16.3i)T + (-802. - 1.66e3i)T^{2} \)
47 \( 1 + (-59.1 - 6.66i)T + (2.15e3 + 491. i)T^{2} \)
53 \( 1 + (-8.89 + 38.9i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 + 6.74T + 3.48e3T^{2} \)
61 \( 1 + (-74.0 - 25.9i)T + (2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (-29.6 - 23.6i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (65.3 - 52.0i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (38.0 + 23.8i)T + (2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (120. - 13.6i)T + (6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (7.07 + 3.40i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (-94.3 + 59.3i)T + (3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (13.0 - 4.56i)T + (7.35e3 - 5.86e3i)T^{2} \)
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

−11.73760013853537406259036157308, −10.68106941765183983125560007015, −10.02840584317016201525227379072, −9.127869801977198623775694952905, −6.95128248933498637061502654848, −5.79311600334122393881970848198, −5.56111782042330169005634353861, −3.79032842926130151151718700986, −2.91369132915964203581959967643, −1.63761581779801626440110252726, 2.52363400680549019153839818441, 3.80145370621614570016578685821, 4.95809090802421723474304551262, 5.93400823248299953746127049832, 6.69120278893525386538964927729, 7.56373426097447831232466962971, 8.988184417110503052031407196091, 10.07668652375935282689303010810, 11.49670748152607380828286166380, 12.47240819189953650322773282278

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