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Properties

Label 2-891-99.31-c1-0-14
Degree $2$
Conductor $891$
Sign $0.0823 - 0.996i$
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  + (0.604 − 0.128i)2-s + (−1.47 + 0.658i)4-s + (2.56 + 0.544i)5-s + (−0.313 + 2.98i)7-s + (−1.80 + 1.31i)8-s + 1.61·10-s + (1.62 − 2.89i)11-s + (1.18 + 1.31i)13-s + (0.193 + 1.84i)14-s + (1.24 − 1.37i)16-s + (0.5 + 1.53i)17-s + (−4.73 + 3.44i)19-s + (−4.14 + 0.880i)20-s + (0.611 − 1.95i)22-s + (−1.73 + 3.00i)23-s + ⋯
L(s)  = 1  + (0.427 − 0.0908i)2-s + (−0.739 + 0.329i)4-s + (1.14 + 0.243i)5-s + (−0.118 + 1.12i)7-s + (−0.639 + 0.464i)8-s + 0.511·10-s + (0.490 − 0.871i)11-s + (0.327 + 0.363i)13-s + (0.0517 + 0.492i)14-s + (0.310 − 0.344i)16-s + (0.121 + 0.373i)17-s + (−1.08 + 0.789i)19-s + (−0.926 + 0.196i)20-s + (0.130 − 0.417i)22-s + (−0.361 + 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0823 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0823 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.0823 - 0.996i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.0823 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28193 + 1.18036i\)
\(L(\frac12)\) \(\approx\) \(1.28193 + 1.18036i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.89i)T \)
good2 \( 1 + (-0.604 + 0.128i)T + (1.82 - 0.813i)T^{2} \)
5 \( 1 + (-2.56 - 0.544i)T + (4.56 + 2.03i)T^{2} \)
7 \( 1 + (0.313 - 2.98i)T + (-6.84 - 1.45i)T^{2} \)
13 \( 1 + (-1.18 - 1.31i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.5 - 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.73 - 3.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.467 - 4.44i)T + (-28.3 - 6.02i)T^{2} \)
31 \( 1 + (-1.90 - 2.12i)T + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.24 - 11.8i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (3.11 + 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.47 - 0.658i)T + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (2.97 - 9.14i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.43 + 4.20i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-5.25 + 5.83i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (-4.78 + 8.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.71 + 5.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (9.26 - 1.96i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (-0.473 + 0.526i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 - 0.527T + 89T^{2} \)
97 \( 1 + (-13.7 + 2.91i)T + (88.6 - 39.4i)T^{2} \)
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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

−10.17660814445836439883335793383, −9.373468554637043962640976364709, −8.752401696849199828363080346334, −8.112795540244595131833746439352, −6.42365176955243901194425651610, −5.95221338099095994270208714673, −5.19743774005341700971722240639, −3.93722384493155457676010991664, −2.97334185865630287011434867998, −1.79046294989799321267323050959, 0.75696715580205815787365810163, 2.21783501077770317595399763054, 3.87752854261540531062013616525, 4.51970531656669100935201374241, 5.49421365302536754484208999825, 6.36262791189708795547700715229, 7.11696155303242962426262399550, 8.425976375614054530535770512407, 9.229728728991864528651788925348, 10.05407035590974659086017635219

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