Properties

Label 2-891-99.16-c1-0-23
Degree $2$
Conductor $891$
Sign $-0.774 + 0.632i$
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  + (−2.56 − 0.544i)2-s + (4.43 + 1.97i)4-s + (−0.604 + 0.128i)5-s + (0.104 + 0.994i)7-s + (−6.04 − 4.39i)8-s + 1.61·10-s + (−1.46 + 2.97i)11-s + (0.157 − 0.175i)13-s + (0.273 − 2.60i)14-s + (6.59 + 7.32i)16-s + (0.354 − 1.08i)17-s + (−4.73 − 3.44i)19-s + (−2.93 − 0.623i)20-s + (5.35 − 6.83i)22-s + (0.118 + 0.204i)23-s + ⋯
L(s)  = 1  + (−1.81 − 0.384i)2-s + (2.21 + 0.987i)4-s + (−0.270 + 0.0574i)5-s + (0.0395 + 0.375i)7-s + (−2.13 − 1.55i)8-s + 0.511·10-s + (−0.440 + 0.897i)11-s + (0.0438 − 0.0486i)13-s + (0.0731 − 0.695i)14-s + (1.64 + 1.83i)16-s + (0.0858 − 0.264i)17-s + (−1.08 − 0.789i)19-s + (−0.656 − 0.139i)20-s + (1.14 − 1.45i)22-s + (0.0246 + 0.0426i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0661605 - 0.185494i\)
\(L(\frac12)\) \(\approx\) \(0.0661605 - 0.185494i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 - 2.97i)T \)
good2 \( 1 + (2.56 + 0.544i)T + (1.82 + 0.813i)T^{2} \)
5 \( 1 + (0.604 - 0.128i)T + (4.56 - 2.03i)T^{2} \)
7 \( 1 + (-0.104 - 0.994i)T + (-6.84 + 1.45i)T^{2} \)
13 \( 1 + (-0.157 + 0.175i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (-0.354 + 1.08i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.73 + 3.44i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.118 - 0.204i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.627 + 5.96i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (4.07 - 4.52i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-5.04 + 3.66i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.0246 + 0.234i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (-3.35 + 5.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.21 + 4.10i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (-0.118 - 0.363i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.74 + 3.00i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (7.73 + 8.59i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (0.927 + 1.60i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.19 - 9.82i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.61 + 3.35i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (10.7 + 2.28i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (0.985 + 1.09i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 - 8.23T + 89T^{2} \)
97 \( 1 + (7.68 + 1.63i)T + (88.6 + 39.4i)T^{2} \)
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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

−9.627616576066521617504832554103, −9.104283098594605303325295749320, −8.244705072796524129396319793375, −7.48973880269251850197351545295, −6.86779804750725893631412581188, −5.67055544907301635554766842951, −4.16207341311858057853286648486, −2.69893299743495323726151888778, −1.89966034359828342775050008332, −0.17862987779690238603539644202, 1.25041368468186580366881875558, 2.59912943156985068644831643402, 4.06873838931094819213177941241, 5.75072976389137648791958387308, 6.33903401947951808786698690668, 7.50224609761752000575945003058, 7.948362869750974069356352030132, 8.740079836078121697841214348987, 9.438159918840341308416680159656, 10.50980483606750214503360329289

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