Properties

Label 2-891-99.41-c1-0-44
Degree $2$
Conductor $891$
Sign $-0.955 - 0.294i$
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  + (0.231 + 0.103i)2-s + (−1.29 − 1.43i)4-s + (−1.18 − 2.66i)5-s + (−0.565 − 2.65i)7-s + (−0.308 − 0.949i)8-s − 0.741i·10-s + (−3.00 − 1.40i)11-s + (2.84 − 0.298i)13-s + (0.143 − 0.674i)14-s + (−0.378 + 3.59i)16-s + (3.60 + 2.62i)17-s + (−1.81 + 0.590i)19-s + (−2.30 + 5.16i)20-s + (−0.550 − 0.635i)22-s + (−0.706 − 0.408i)23-s + ⋯
L(s)  = 1  + (0.163 + 0.0729i)2-s + (−0.647 − 0.719i)4-s + (−0.531 − 1.19i)5-s + (−0.213 − 1.00i)7-s + (−0.109 − 0.335i)8-s − 0.234i·10-s + (−0.905 − 0.424i)11-s + (0.787 − 0.0827i)13-s + (0.0383 − 0.180i)14-s + (−0.0945 + 0.899i)16-s + (0.875 + 0.636i)17-s + (−0.416 + 0.135i)19-s + (−0.514 + 1.15i)20-s + (−0.117 − 0.135i)22-s + (−0.147 − 0.0851i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.955 - 0.294i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.955 - 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0972286 + 0.646327i\)
\(L(\frac12)\) \(\approx\) \(0.0972286 + 0.646327i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.00 + 1.40i)T \)
good2 \( 1 + (-0.231 - 0.103i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (1.18 + 2.66i)T + (-3.34 + 3.71i)T^{2} \)
7 \( 1 + (0.565 + 2.65i)T + (-6.39 + 2.84i)T^{2} \)
13 \( 1 + (-2.84 + 0.298i)T + (12.7 - 2.70i)T^{2} \)
17 \( 1 + (-3.60 - 2.62i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.81 - 0.590i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.706 + 0.408i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (9.33 - 1.98i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (-0.625 - 5.95i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-1.83 + 5.65i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (8.24 + 1.75i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-10.2 + 5.93i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.73 - 5.16i)T + (4.91 + 46.7i)T^{2} \)
53 \( 1 + (6.14 + 8.45i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0693 - 0.0624i)T + (6.16 - 58.6i)T^{2} \)
61 \( 1 + (2.13 + 0.224i)T + (59.6 + 12.6i)T^{2} \)
67 \( 1 + (-0.703 + 1.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.12 - 2.91i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.67 + 0.543i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.90 - 6.51i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (0.818 - 7.79i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 2.06iT - 89T^{2} \)
97 \( 1 + (-1.52 - 0.679i)T + (64.9 + 72.0i)T^{2} \)
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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

−9.650401024551884382808425093210, −8.729493061578254385725906704091, −8.172689701222563667864619560216, −7.18482643190401077802318908829, −5.85152022545984365708395868628, −5.29123048691258083510101585324, −4.20741562100277371176589446367, −3.62138744165409472072708402422, −1.40399227507258637480364010876, −0.31806722666833370474421879319, 2.48328420848882503180309473298, 3.18779189281510944012807701504, 4.14427533151493394405513533189, 5.34128351287204688058436046368, 6.21935652924674064984685130995, 7.49732314226150668011658311673, 7.83623656076379394965590292621, 8.927794579156231833901332977565, 9.661017160813798636659030095192, 10.66821099897464120624530099749

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