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Properties

Label 2-891-99.58-c1-0-45
Degree $2$
Conductor $891$
Sign $-0.999 + 0.00433i$
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  + (−0.288 − 0.128i)2-s + (−1.27 − 1.41i)4-s + (3.70 − 1.64i)5-s + (−3.33 + 0.709i)7-s + (0.380 + 1.17i)8-s − 1.28·10-s + (−3.27 + 0.553i)11-s + (−0.144 − 1.37i)13-s + (1.05 + 0.224i)14-s + (−0.356 + 3.39i)16-s + (−5.20 − 3.78i)17-s + (−1.07 − 3.31i)19-s + (−7.03 − 3.13i)20-s + (1.01 + 0.260i)22-s + (−1.86 + 3.23i)23-s + ⋯
L(s)  = 1  + (−0.204 − 0.0909i)2-s + (−0.635 − 0.706i)4-s + (1.65 − 0.737i)5-s + (−1.26 + 0.267i)7-s + (0.134 + 0.414i)8-s − 0.405·10-s + (−0.985 + 0.166i)11-s + (−0.0400 − 0.381i)13-s + (0.281 + 0.0599i)14-s + (−0.0891 + 0.847i)16-s + (−1.26 − 0.916i)17-s + (−0.247 − 0.760i)19-s + (−1.57 − 0.700i)20-s + (0.216 + 0.0555i)22-s + (−0.389 + 0.675i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $-0.999 + 0.00433i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.999 + 0.00433i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00115226 - 0.531559i\)
\(L(\frac12)\) \(\approx\) \(0.00115226 - 0.531559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.27 - 0.553i)T \)
good2 \( 1 + (0.288 + 0.128i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (-3.70 + 1.64i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (3.33 - 0.709i)T + (6.39 - 2.84i)T^{2} \)
13 \( 1 + (0.144 + 1.37i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (5.20 + 3.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.07 + 3.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.86 - 3.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0729 + 0.0155i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (-0.390 - 3.71i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (0.381 - 1.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (9.11 + 1.93i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-0.228 - 0.395i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.46 - 1.63i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (-0.729 + 0.530i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (5.36 + 5.96i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.603 - 5.74i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (-2.92 + 5.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.66 - 5.56i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.34 + 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.0 + 4.47i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-0.143 + 1.36i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (7.53 + 3.35i)T + (64.9 + 72.0i)T^{2} \)
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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

−9.694558936970992613104005823819, −9.127802581382164894047161295964, −8.457855380942469724906773910636, −6.86614908218732695497382061168, −6.08063363506701710749316199875, −5.29444794265520374157331139724, −4.71998437767867085207444612431, −2.87604010199922948996205297448, −1.87107594056148905012902826354, −0.25106705897357914654426306222, 2.14522994141274730155497868810, 3.08028494503808089961125791481, 4.15890571420937113982065556049, 5.49334389544747919111213988714, 6.41282177375271595741202485169, 6.88673982472043462244172891557, 8.157172991634048557322116524693, 8.976392102205863571291652830059, 9.884082252155495719460718097397, 10.14902832145782431836813511781

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