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Properties

Label 2-891-99.50-c1-0-6
Degree $2$
Conductor $891$
Sign $-0.956 + 0.290i$
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  + (−1.57 + 1.74i)2-s + (−0.368 − 3.50i)4-s + (1.67 − 1.50i)5-s + (−1.64 + 3.70i)7-s + (2.90 + 2.10i)8-s + 5.28i·10-s + (−1.64 − 2.88i)11-s + (−0.967 + 4.55i)13-s + (−3.87 − 8.69i)14-s + (−1.35 + 0.287i)16-s + (0.0235 − 0.0725i)17-s + (1.40 − 1.93i)19-s + (−5.89 − 5.30i)20-s + (7.61 + 1.66i)22-s + (2.79 + 1.61i)23-s + ⋯
L(s)  = 1  + (−1.11 + 1.23i)2-s + (−0.184 − 1.75i)4-s + (0.747 − 0.672i)5-s + (−0.622 + 1.39i)7-s + (1.02 + 0.745i)8-s + 1.67i·10-s + (−0.494 − 0.869i)11-s + (−0.268 + 1.26i)13-s + (−1.03 − 2.32i)14-s + (−0.337 + 0.0718i)16-s + (0.00571 − 0.0175i)17-s + (0.323 − 0.444i)19-s + (−1.31 − 1.18i)20-s + (1.62 + 0.355i)22-s + (0.582 + 0.336i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.956 + 0.290i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0677313 - 0.455560i\)
\(L(\frac12)\) \(\approx\) \(0.0677313 - 0.455560i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 2.88i)T \)
good2 \( 1 + (1.57 - 1.74i)T + (-0.209 - 1.98i)T^{2} \)
5 \( 1 + (-1.67 + 1.50i)T + (0.522 - 4.97i)T^{2} \)
7 \( 1 + (1.64 - 3.70i)T + (-4.68 - 5.20i)T^{2} \)
13 \( 1 + (0.967 - 4.55i)T + (-11.8 - 5.28i)T^{2} \)
17 \( 1 + (-0.0235 + 0.0725i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.40 + 1.93i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.79 - 1.61i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.68 - 0.748i)T + (19.4 + 21.5i)T^{2} \)
31 \( 1 + (1.63 + 0.348i)T + (28.3 + 12.6i)T^{2} \)
37 \( 1 + (5.87 - 4.27i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.71 - 3.43i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (3.70 - 2.14i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.18 + 0.649i)T + (45.9 + 9.77i)T^{2} \)
53 \( 1 + (1.16 - 0.379i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.577 - 0.0606i)T + (57.7 - 12.2i)T^{2} \)
61 \( 1 + (-0.786 - 3.69i)T + (-55.7 + 24.8i)T^{2} \)
67 \( 1 + (6.49 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.06 + 0.346i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.82 - 10.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.490 + 0.441i)T + (8.25 + 78.5i)T^{2} \)
83 \( 1 + (9.98 - 2.12i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 - 6.58iT - 89T^{2} \)
97 \( 1 + (-11.0 + 12.2i)T + (-10.1 - 96.4i)T^{2} \)
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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

−9.924249592480544001671108116429, −9.507301539866180420947150320993, −8.713200003758336213774485925009, −8.476067465430572724646513663358, −7.09628218643863951836869574004, −6.40294935672765647984468902465, −5.57564462818091248439986463738, −5.04660873802480792559478923683, −3.00446065400592235486454900615, −1.55282214765201905211092631336, 0.31581607108370664396455924587, 1.78073586988085777133816918406, 2.91095742900306086839346262008, 3.63982106694212213413156551447, 5.11946983559820048710667535912, 6.51222176396581372072217994136, 7.37478986848033725397834451865, 8.044170526332368364459707185820, 9.249169374309787115887109289271, 10.04449706452623660807299612516

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