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Properties

Label 2-891-99.4-c1-0-37
Degree $2$
Conductor $891$
Sign $0.797 + 0.602i$
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  + (−0.120 + 1.14i)2-s + (0.662 + 0.140i)4-s + (−0.408 − 3.88i)5-s + (3.06 − 3.40i)7-s + (−0.951 + 2.92i)8-s + 4.49·10-s + (−3.04 + 1.32i)11-s + (−0.533 + 0.237i)13-s + (3.53 + 3.92i)14-s + (−1.99 − 0.889i)16-s + (1.01 − 0.736i)17-s + (−0.681 + 2.09i)19-s + (0.276 − 2.63i)20-s + (−1.15 − 3.63i)22-s + (3.65 − 6.32i)23-s + ⋯
L(s)  = 1  + (−0.0850 + 0.808i)2-s + (0.331 + 0.0704i)4-s + (−0.182 − 1.73i)5-s + (1.16 − 1.28i)7-s + (−0.336 + 1.03i)8-s + 1.42·10-s + (−0.916 + 0.399i)11-s + (−0.147 + 0.0658i)13-s + (0.943 + 1.04i)14-s + (−0.499 − 0.222i)16-s + (0.245 − 0.178i)17-s + (−0.156 + 0.481i)19-s + (0.0619 − 0.589i)20-s + (−0.245 − 0.775i)22-s + (0.761 − 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.797 + 0.602i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.797 + 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60681 - 0.538839i\)
\(L(\frac12)\) \(\approx\) \(1.60681 - 0.538839i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 - 1.32i)T \)
good2 \( 1 + (0.120 - 1.14i)T + (-1.95 - 0.415i)T^{2} \)
5 \( 1 + (0.408 + 3.88i)T + (-4.89 + 1.03i)T^{2} \)
7 \( 1 + (-3.06 + 3.40i)T + (-0.731 - 6.96i)T^{2} \)
13 \( 1 + (0.533 - 0.237i)T + (8.69 - 9.66i)T^{2} \)
17 \( 1 + (-1.01 + 0.736i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.681 - 2.09i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.65 + 6.32i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.832 + 0.924i)T + (-3.03 - 28.8i)T^{2} \)
31 \( 1 + (-7.92 + 3.52i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (2.36 + 7.26i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.10 + 1.22i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (2.89 + 5.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.516 - 0.109i)T + (42.9 - 19.1i)T^{2} \)
53 \( 1 + (1.07 + 0.782i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.543 + 0.115i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-10.2 - 4.55i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (3.73 - 6.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.14 + 3.00i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.117 - 0.361i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.42 - 13.5i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (-0.912 - 0.406i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 - 3.33T + 89T^{2} \)
97 \( 1 + (1.64 - 15.6i)T + (-94.8 - 20.1i)T^{2} \)
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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

−10.04786092611326164043988473712, −8.763052374080551201244359445474, −8.158311193521608164421331967942, −7.69321922819917590812101720809, −6.81976160819412178079430825706, −5.45505434194267567824241007910, −4.88289187216579821036327320991, −4.13495362934067372893914568036, −2.18920080349328549357891927547, −0.840337189161243563326908813157, 1.76830143833905369756116534044, 2.81870094865350572881589750434, 3.17749633794413612679445715085, 4.91377145411466062937528497666, 5.93863188774478899874855599763, 6.78881280794298169673014002806, 7.69113504935028577293387514964, 8.497434035106849201232143485131, 9.734081451471609600685279907171, 10.43038561129589446867503529082

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