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Properties

Label 2-891-99.58-c1-0-2
Degree $2$
Conductor $891$
Sign $0.126 - 0.992i$
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  + (−1.47 − 0.658i)2-s + (0.413 + 0.459i)4-s + (0.348 − 0.155i)5-s + (−2.93 + 0.623i)7-s + (0.690 + 2.12i)8-s − 0.618·10-s + (2.93 + 1.55i)11-s + (−0.651 − 6.20i)13-s + (4.74 + 1.00i)14-s + (0.507 − 4.82i)16-s + (−0.5 − 0.363i)17-s + (−0.263 − 0.812i)19-s + (0.215 + 0.0960i)20-s + (−3.31 − 4.22i)22-s + (−2.73 + 4.73i)23-s + ⋯
L(s)  = 1  + (−1.04 − 0.465i)2-s + (0.206 + 0.229i)4-s + (0.156 − 0.0694i)5-s + (−1.10 + 0.235i)7-s + (0.244 + 0.751i)8-s − 0.195·10-s + (0.883 + 0.467i)11-s + (−0.180 − 1.72i)13-s + (1.26 + 0.269i)14-s + (0.126 − 1.20i)16-s + (−0.121 − 0.0881i)17-s + (−0.0605 − 0.186i)19-s + (0.0482 + 0.0214i)20-s + (−0.706 − 0.900i)22-s + (−0.570 + 0.988i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.126 - 0.992i$
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: \((2,\ 891,\ (\ :1/2),\ 0.126 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257578 + 0.226911i\)
\(L(\frac12)\) \(\approx\) \(0.257578 + 0.226911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.93 - 1.55i)T \)
good2 \( 1 + (1.47 + 0.658i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (-0.348 + 0.155i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (2.93 - 0.623i)T + (6.39 - 2.84i)T^{2} \)
13 \( 1 + (0.651 + 6.20i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.73 - 4.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.37 - 0.929i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (-0.402 - 3.83i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (1.30 - 4.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-5.81 - 1.23i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (0.881 + 1.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.413 + 0.459i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (5.97 - 4.33i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.56 - 3.95i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.119 - 1.13i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (5.28 - 9.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.7 - 8.55i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.381 + 1.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.482 - 0.214i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (1.32 - 12.6i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 9.47T + 89T^{2} \)
97 \( 1 + (-13.7 - 6.11i)T + (64.9 + 72.0i)T^{2} \)
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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

−9.990987059363756464581337544441, −9.630593486564976684089781656811, −8.928613002717372904191370116655, −7.958707104356809668026152280548, −7.15178084187197774759036700699, −5.96862593014312081620694290270, −5.21520576199676047083684265691, −3.69547561207151089906393923575, −2.64753482518999697368200699475, −1.28340068937327473658277370192, 0.25140674087171197449900242743, 1.96949444662861861830744053029, 3.66975999772290004218645701186, 4.31722053783919289054556250604, 6.26120616843810326914977557664, 6.44559692985858460970572998088, 7.40798867605067531495706292028, 8.364853839541820888076448552797, 9.353783196980864649615716938434, 9.462380822843762222824856273671

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