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Properties

Label 2-891-99.49-c1-0-38
Degree $2$
Conductor $891$
Sign $-0.577 + 0.816i$
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  + (−0.413 + 0.459i)2-s + (0.169 + 1.60i)4-s + (−1.75 − 1.94i)5-s + (2.74 + 1.22i)7-s + (−1.80 − 1.31i)8-s + 1.61·10-s + (−3.31 − 0.0378i)11-s + (−1.72 − 0.366i)13-s + (−1.69 + 0.754i)14-s + (−1.81 + 0.385i)16-s + (0.5 − 1.53i)17-s + (−4.73 − 3.44i)19-s + (2.83 − 3.14i)20-s + (1.38 − 1.50i)22-s + (−1.73 + 3.00i)23-s + ⋯
L(s)  = 1  + (−0.292 + 0.324i)2-s + (0.0845 + 0.804i)4-s + (−0.783 − 0.870i)5-s + (1.03 + 0.461i)7-s + (−0.639 − 0.464i)8-s + 0.511·10-s + (−0.999 − 0.0114i)11-s + (−0.478 − 0.101i)13-s + (−0.452 + 0.201i)14-s + (−0.453 + 0.0963i)16-s + (0.121 − 0.373i)17-s + (−1.08 − 0.789i)19-s + (0.633 − 0.703i)20-s + (0.296 − 0.321i)22-s + (−0.361 + 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0916411 - 0.177079i\)
\(L(\frac12)\) \(\approx\) \(0.0916411 - 0.177079i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 0.0378i)T \)
good2 \( 1 + (0.413 - 0.459i)T + (-0.209 - 1.98i)T^{2} \)
5 \( 1 + (1.75 + 1.94i)T + (-0.522 + 4.97i)T^{2} \)
7 \( 1 + (-2.74 - 1.22i)T + (4.68 + 5.20i)T^{2} \)
13 \( 1 + (1.72 + 0.366i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (-0.5 + 1.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.73 + 3.44i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.73 - 3.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.08 - 1.81i)T + (19.4 + 21.5i)T^{2} \)
31 \( 1 + (2.79 + 0.593i)T + (28.3 + 12.6i)T^{2} \)
37 \( 1 + (0.190 - 0.138i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (10.9 - 4.85i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (3.11 + 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.169 - 1.60i)T + (-45.9 - 9.77i)T^{2} \)
53 \( 1 + (2.97 + 9.14i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.07 + 10.2i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + (7.68 - 1.63i)T + (55.7 - 24.8i)T^{2} \)
67 \( 1 + (-4.78 + 8.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.71 - 5.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.61 + 1.90i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-6.33 + 7.03i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (0.692 - 0.147i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 - 0.527T + 89T^{2} \)
97 \( 1 + (9.39 - 10.4i)T + (-10.1 - 96.4i)T^{2} \)
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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

−9.583045780717151366373077394098, −8.567582584816788377212023668987, −8.239519955814472085550976227822, −7.60690586665240071533010670225, −6.61543401195706997478151145431, −5.12285455618462304151794068437, −4.68470661094258659774781300409, −3.41037307289947161918938431501, −2.14806709587009655147082975173, −0.099245298131578317865140551183, 1.70679342235085118965729824256, 2.80322057047166928682795845695, 4.16986916589665293680428434535, 5.06632658723388647345000050588, 6.14619260568166271869502341942, 7.11466458655602225577463591479, 7.969118327762431040530148159161, 8.583099688700708566417150750593, 9.964787034228389846394675532973, 10.56299705212249245130226827318

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