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Properties

Label 2-891-33.2-c1-0-22
Degree $2$
Conductor $891$
Sign $0.490 + 0.871i$
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  + (0.525 − 1.61i)2-s + (−0.722 − 0.525i)4-s + (1.67 − 0.543i)5-s + (−1.64 + 2.27i)7-s + (1.52 − 1.10i)8-s − 2.99i·10-s + (3.04 + 1.31i)11-s + (3.13 + 1.01i)13-s + (2.80 + 3.86i)14-s + (−1.54 − 4.74i)16-s + (2.08 + 6.41i)17-s + (0.847 + 1.16i)19-s + (−1.49 − 0.485i)20-s + (3.72 − 4.23i)22-s − 1.74i·23-s + ⋯
L(s)  = 1  + (0.371 − 1.14i)2-s + (−0.361 − 0.262i)4-s + (0.748 − 0.243i)5-s + (−0.623 + 0.858i)7-s + (0.538 − 0.391i)8-s − 0.946i·10-s + (0.917 + 0.396i)11-s + (0.869 + 0.282i)13-s + (0.750 + 1.03i)14-s + (−0.385 − 1.18i)16-s + (0.505 + 1.55i)17-s + (0.194 + 0.267i)19-s + (−0.334 − 0.108i)20-s + (0.795 − 0.902i)22-s − 0.364i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.490 + 0.871i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (728, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.490 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08623 - 1.22013i\)
\(L(\frac12)\) \(\approx\) \(2.08623 - 1.22013i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 - 1.31i)T \)
good2 \( 1 + (-0.525 + 1.61i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.67 + 0.543i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.64 - 2.27i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-3.13 - 1.01i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.08 - 6.41i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.847 - 1.16i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.74iT - 23T^{2} \)
29 \( 1 + (4.79 + 3.48i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.50 + 4.63i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.98 + 2.16i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.61 + 1.89i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.71iT - 43T^{2} \)
47 \( 1 + (-3.45 - 4.74i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.88 - 1.26i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.41 + 1.94i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.42 + 1.76i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + (7.80 - 2.53i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.29 + 7.28i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (13.7 + 4.46i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.308 - 0.949i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 14.0iT - 89T^{2} \)
97 \( 1 + (3.86 - 11.8i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.06130664882060484796782792644, −9.371637797897214741687642752022, −8.685012305584136461247751258238, −7.39885567290126737640583686026, −6.16724805468267652907960887176, −5.73200088706256936582519908719, −4.19360042034405821133371571233, −3.55536033767171675317953977969, −2.24559591472535181782082239409, −1.48041742531839681198953362507, 1.28636806435804857279145406620, 3.05035876034777515278211992678, 4.12171271356274624184421193034, 5.32970490678453519852501046867, 6.01347905785875084302424627515, 6.85449166060967698632671056149, 7.28509676920366135474371595073, 8.446488668705366950451113128067, 9.387102891518592696257150715102, 10.15199488199463232397592859972

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