Properties

Label 2-891-11.9-c1-0-34
Degree $2$
Conductor $891$
Sign $0.244 + 0.969i$
Analytic cond. $7.11467$
Root an. cond. $2.66733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.143 − 0.441i)2-s + (1.44 + 1.04i)4-s + (−0.808 − 2.48i)5-s + (1.43 + 1.04i)7-s + (1.42 − 1.03i)8-s − 1.21·10-s + (−1.63 − 2.88i)11-s + (1.67 − 5.14i)13-s + (0.667 − 0.484i)14-s + (0.851 + 2.62i)16-s + (−0.343 − 1.05i)17-s + (−2.64 + 1.92i)19-s + (1.44 − 4.44i)20-s + (−1.50 + 0.308i)22-s − 4.23·23-s + ⋯
L(s)  = 1  + (0.101 − 0.311i)2-s + (0.721 + 0.524i)4-s + (−0.361 − 1.11i)5-s + (0.543 + 0.394i)7-s + (0.502 − 0.364i)8-s − 0.383·10-s + (−0.493 − 0.869i)11-s + (0.463 − 1.42i)13-s + (0.178 − 0.129i)14-s + (0.212 + 0.655i)16-s + (−0.0832 − 0.256i)17-s + (−0.606 + 0.440i)19-s + (0.322 − 0.993i)20-s + (−0.321 + 0.0657i)22-s − 0.883·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.244 + 0.969i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.244 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50293 - 1.17042i\)
\(L(\frac12)\) \(\approx\) \(1.50293 - 1.17042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (1.63 + 2.88i)T \)
good2 \( 1 + (-0.143 + 0.441i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (0.808 + 2.48i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.43 - 1.04i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.67 + 5.14i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.343 + 1.05i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.64 - 1.92i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.23T + 23T^{2} \)
29 \( 1 + (-6.27 - 4.55i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.95 + 9.08i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.95 - 2.87i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.12 + 2.99i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.53T + 43T^{2} \)
47 \( 1 + (-3.69 + 2.68i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.06 - 9.44i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.48 + 1.80i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.65 - 5.08i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 9.10T + 67T^{2} \)
71 \( 1 + (-0.243 - 0.749i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.16 - 0.847i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.23 + 6.87i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.53 + 10.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (5.82 - 17.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.21035922364823333060670477048, −8.828121790088644745320986108429, −8.021283721156084976844330867326, −7.965371006475161598444191154040, −6.35911670695631137292994459865, −5.52956163254813010787668258379, −4.51956647593784541243606793821, −3.45165180459669602374737178128, −2.38476828721677563952413160816, −0.913729455075571771908963444470, 1.70783252215866727666049649782, 2.68320422680292655670745734750, 4.14001904024897301673368924009, 4.95471437242396412617953316929, 6.45895905641693791547629001900, 6.65739012763809559429570845266, 7.57543349407537383520085700121, 8.352227394979871301472429451649, 9.736345272333915980987762917453, 10.41965692948420846352930460946

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