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Properties

Label 2-891-99.2-c1-0-41
Degree $2$
Conductor $891$
Sign $0.456 + 0.889i$
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  + (0.530 + 0.589i)2-s + (0.143 − 1.36i)4-s + (2.46 + 2.21i)5-s + (−1.94 − 4.35i)7-s + (2.16 − 1.57i)8-s + 2.62i·10-s + (−2.32 − 2.36i)11-s + (−0.685 − 3.22i)13-s + (1.53 − 3.45i)14-s + (−0.610 − 0.129i)16-s + (0.204 + 0.628i)17-s + (−2.30 − 3.17i)19-s + (3.37 − 3.04i)20-s + (0.164 − 2.62i)22-s + (−2.58 + 1.49i)23-s + ⋯
L(s)  = 1  + (0.375 + 0.416i)2-s + (0.0716 − 0.682i)4-s + (1.10 + 0.992i)5-s + (−0.733 − 1.64i)7-s + (0.764 − 0.555i)8-s + 0.831i·10-s + (−0.699 − 0.714i)11-s + (−0.190 − 0.894i)13-s + (0.411 − 0.923i)14-s + (−0.152 − 0.0324i)16-s + (0.0495 + 0.152i)17-s + (−0.528 − 0.727i)19-s + (0.755 − 0.680i)20-s + (0.0351 − 0.559i)22-s + (−0.539 + 0.311i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.456 + 0.889i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65806 - 1.01317i\)
\(L(\frac12)\) \(\approx\) \(1.65806 - 1.01317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.32 + 2.36i)T \)
good2 \( 1 + (-0.530 - 0.589i)T + (-0.209 + 1.98i)T^{2} \)
5 \( 1 + (-2.46 - 2.21i)T + (0.522 + 4.97i)T^{2} \)
7 \( 1 + (1.94 + 4.35i)T + (-4.68 + 5.20i)T^{2} \)
13 \( 1 + (0.685 + 3.22i)T + (-11.8 + 5.28i)T^{2} \)
17 \( 1 + (-0.204 - 0.628i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.30 + 3.17i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.58 - 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.46 - 0.652i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (-1.72 + 0.367i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-7.22 - 5.24i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.07 - 1.37i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (-6.23 - 3.60i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.26 - 0.763i)T + (45.9 - 9.77i)T^{2} \)
53 \( 1 + (-2.84 - 0.923i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-13.7 - 1.44i)T + (57.7 + 12.2i)T^{2} \)
61 \( 1 + (0.200 - 0.944i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (-0.535 - 0.927i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-14.2 + 4.62i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.33 + 7.33i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.76 + 5.19i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (-5.17 - 1.09i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 - 3.98iT - 89T^{2} \)
97 \( 1 + (2.46 + 2.73i)T + (-10.1 + 96.4i)T^{2} \)
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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

−9.956508989006961157456895990086, −9.675088527345566173238128019403, −7.973088566807308044924734112400, −7.14799785027602773428042757323, −6.38809012068360910791385637074, −5.89869806002243345789103292304, −4.79367615208407279895071175448, −3.57177025602280250252447052342, −2.49200375602354746936441767563, −0.794733591620180940668248858697, 2.15177009244002584168835809390, 2.36749915678658238882169767770, 3.96732872625861383176812470361, 5.03208819742032837727714507666, 5.70803537686172485795835026912, 6.64848325647315316148730106311, 7.964766263822044644593996769898, 8.745552777279816320173290786879, 9.433346794596798291508598967049, 10.03896834558561032386605040050

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