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Properties

Label 2-891-99.58-c1-0-35
Degree $2$
Conductor $891$
Sign $0.716 + 0.697i$
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  + (0.974 + 0.433i)2-s + (−0.576 − 0.640i)4-s + (1.12 − 0.499i)5-s + (1.49 − 0.318i)7-s + (−0.943 − 2.90i)8-s + 1.30·10-s + (2.72 + 1.89i)11-s + (−0.244 − 2.32i)13-s + (1.59 + 0.339i)14-s + (0.160 − 1.52i)16-s + (−2.87 − 2.08i)17-s + (−0.884 − 2.72i)19-s + (−0.966 − 0.430i)20-s + (1.82 + 3.02i)22-s + (1.14 − 1.98i)23-s + ⋯
L(s)  = 1  + (0.689 + 0.306i)2-s + (−0.288 − 0.320i)4-s + (0.501 − 0.223i)5-s + (0.566 − 0.120i)7-s + (−0.333 − 1.02i)8-s + 0.414·10-s + (0.820 + 0.572i)11-s + (−0.0679 − 0.646i)13-s + (0.427 + 0.0908i)14-s + (0.0400 − 0.381i)16-s + (−0.696 − 0.506i)17-s + (−0.202 − 0.624i)19-s + (−0.216 − 0.0962i)20-s + (0.389 + 0.645i)22-s + (0.239 − 0.414i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.716 + 0.697i$
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: \((2,\ 891,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11242 - 0.859049i\)
\(L(\frac12)\) \(\approx\) \(2.11242 - 0.859049i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.72 - 1.89i)T \)
good2 \( 1 + (-0.974 - 0.433i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (-1.12 + 0.499i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (-1.49 + 0.318i)T + (6.39 - 2.84i)T^{2} \)
13 \( 1 + (0.244 + 2.32i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (2.87 + 2.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.884 + 2.72i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.14 + 1.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.11 + 1.72i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (0.505 + 4.80i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-1.24 + 3.82i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-4.73 - 1.00i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-3.76 - 6.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.12 + 6.80i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (-5.00 + 3.63i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (7.28 + 8.08i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-0.0640 + 0.609i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (4.32 - 7.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.6 - 9.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.68 - 14.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.77 + 0.790i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (0.557 - 5.30i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 5.84T + 89T^{2} \)
97 \( 1 + (7.77 + 3.45i)T + (64.9 + 72.0i)T^{2} \)
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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

−9.833387980104360335133228636373, −9.332345432413729277367406764139, −8.403551672042432208791939079406, −7.21781512034309408338047879183, −6.43008627294882069392059902601, −5.54409870607725297328626686888, −4.69947825168075229869330854296, −4.05240020892792823362475509202, −2.48017424512697131840371097180, −0.968451488520636708390312717004, 1.69647542572714314624934321096, 2.86671192688035920406436683654, 4.00143713754086250437588171276, 4.69371661615430367133557975506, 5.82361011866541695973912713354, 6.53498662554554933158468372860, 7.79388356318053767931371914380, 8.697489465414811613131089771127, 9.188725933729856368595379060438, 10.44555444551924651203937173190

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