Properties

Label 2-261-29.27-c2-0-7
Degree $2$
Conductor $261$
Sign $-0.966 - 0.256i$
Analytic cond. $7.11173$
Root an. cond. $2.66678$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.293 + 2.60i)2-s + (−2.78 − 0.635i)4-s + (1.65 − 1.31i)5-s + (0.747 + 3.27i)7-s + (−0.990 + 2.83i)8-s + (2.94 + 4.68i)10-s + (0.536 + 1.53i)11-s + (−3.68 + 7.65i)13-s + (−8.74 + 0.985i)14-s + (−17.3 − 8.36i)16-s + (19.7 + 19.7i)17-s + (−9.28 + 5.83i)19-s + (−5.43 + 2.61i)20-s + (−4.14 + 0.946i)22-s + (−20.8 + 26.1i)23-s + ⋯
L(s)  = 1  + (−0.146 + 1.30i)2-s + (−0.695 − 0.158i)4-s + (0.330 − 0.263i)5-s + (0.106 + 0.468i)7-s + (−0.123 + 0.353i)8-s + (0.294 + 0.468i)10-s + (0.0487 + 0.139i)11-s + (−0.283 + 0.589i)13-s + (−0.624 + 0.0703i)14-s + (−1.08 − 0.522i)16-s + (1.16 + 1.16i)17-s + (−0.488 + 0.307i)19-s + (−0.271 + 0.130i)20-s + (−0.188 + 0.0430i)22-s + (−0.907 + 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(7.11173\)
Root analytic conductor: \(2.66678\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1),\ -0.966 - 0.256i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.178598 + 1.36771i\)
\(L(\frac12)\) \(\approx\) \(0.178598 + 1.36771i\)
\(L(2)\) 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 \( 1 + (12.6 + 26.0i)T \)
good2 \( 1 + (0.293 - 2.60i)T + (-3.89 - 0.890i)T^{2} \)
5 \( 1 + (-1.65 + 1.31i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (-0.747 - 3.27i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (-0.536 - 1.53i)T + (-94.6 + 75.4i)T^{2} \)
13 \( 1 + (3.68 - 7.65i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (-19.7 - 19.7i)T + 289iT^{2} \)
19 \( 1 + (9.28 - 5.83i)T + (156. - 325. i)T^{2} \)
23 \( 1 + (20.8 - 26.1i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (-3.96 + 35.1i)T + (-936. - 213. i)T^{2} \)
37 \( 1 + (-2.28 + 6.52i)T + (-1.07e3 - 853. i)T^{2} \)
41 \( 1 + (-22.9 + 22.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (-55.6 + 6.26i)T + (1.80e3 - 411. i)T^{2} \)
47 \( 1 + (60.5 - 21.2i)T + (1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-22.4 - 28.0i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 - 84.6T + 3.48e3T^{2} \)
61 \( 1 + (38.2 - 60.8i)T + (-1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (14.4 + 29.9i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (-2.14 + 4.45i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (-8.36 - 74.2i)T + (-5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (-28.5 - 9.99i)T + (4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (-7.54 + 33.0i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (7.04 - 62.4i)T + (-7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (21.4 + 34.0i)T + (-4.08e3 + 8.47e3i)T^{2} \)
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

−12.21797933904875156404828143599, −11.36117346137813246747475627091, −9.929401924550696046672641475412, −9.115595638160801462041359589877, −8.073590907336909171344773804295, −7.38184576449500737522260666038, −5.97912189220028934077219232346, −5.61920257902898335732616979432, −4.07045765452900922980634648207, −2.03244780873051719598311409702, 0.74407171575443504039284633878, 2.36431083009656460350419908041, 3.42053673841363778312409984157, 4.78603474352552277390365427493, 6.27959138841627373657489000737, 7.42693672815868866497058351994, 8.709283296317209788837238973353, 9.900660746103112412160374108120, 10.34341115920334638312932639982, 11.23383138211502055542656600170

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