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Properties

Label 2-891-99.83-c1-0-33
Degree $2$
Conductor $891$
Sign $0.545 - 0.837i$
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  + (2.29 + 0.488i)2-s + (3.22 + 1.43i)4-s + (0.467 + 2.19i)5-s + (4.02 − 0.423i)7-s + (2.90 + 2.10i)8-s + 5.28i·10-s + (−1.67 + 2.86i)11-s + (−3.45 − 3.11i)13-s + (9.47 + 0.995i)14-s + (0.924 + 1.02i)16-s + (0.0235 − 0.0725i)17-s + (1.40 − 1.93i)19-s + (−1.64 + 7.75i)20-s + (−5.25 + 5.76i)22-s + (−2.79 + 1.61i)23-s + ⋯
L(s)  = 1  + (1.62 + 0.345i)2-s + (1.61 + 0.717i)4-s + (0.209 + 0.983i)5-s + (1.52 − 0.160i)7-s + (1.02 + 0.745i)8-s + 1.67i·10-s + (−0.505 + 0.862i)11-s + (−0.958 − 0.863i)13-s + (2.53 + 0.266i)14-s + (0.231 + 0.256i)16-s + (0.00571 − 0.0175i)17-s + (0.323 − 0.444i)19-s + (−0.368 + 1.73i)20-s + (−1.11 + 1.22i)22-s + (−0.582 + 0.336i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.545 - 0.837i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.545 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.81177 + 2.06589i\)
\(L(\frac12)\) \(\approx\) \(3.81177 + 2.06589i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 2.86i)T \)
good2 \( 1 + (-2.29 - 0.488i)T + (1.82 + 0.813i)T^{2} \)
5 \( 1 + (-0.467 - 2.19i)T + (-4.56 + 2.03i)T^{2} \)
7 \( 1 + (-4.02 + 0.423i)T + (6.84 - 1.45i)T^{2} \)
13 \( 1 + (3.45 + 3.11i)T + (1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.0235 + 0.0725i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.40 + 1.93i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.79 - 1.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.192 + 1.82i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-1.12 + 1.24i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (5.87 - 4.27i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.882 + 8.39i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (-3.70 - 2.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.52 - 5.67i)T + (-31.4 + 34.9i)T^{2} \)
53 \( 1 + (1.16 - 0.379i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.236 + 0.530i)T + (-39.4 - 43.8i)T^{2} \)
61 \( 1 + (-2.81 + 2.53i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (6.49 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.06 + 0.346i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.82 - 10.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.137 - 0.645i)T + (-72.1 - 32.1i)T^{2} \)
83 \( 1 + (-6.83 - 7.58i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 - 6.58iT - 89T^{2} \)
97 \( 1 + (16.0 + 3.41i)T + (88.6 + 39.4i)T^{2} \)
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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

−10.60241872639528209834307777614, −9.653385749913679221553885050492, −8.041702387740901022649506732614, −7.44477524204000448192110615180, −6.77827428992710736379676883829, −5.57576157318663558242918343313, −5.00504405707969352901762089459, −4.20234992092949801045377528837, −2.92004707874425307027636419572, −2.13236786895355417796477704390, 1.52192033214659596646288160902, 2.49354918357770145225160778316, 3.89155765119765608538010990841, 4.84282518320956400876577001743, 5.16702345689764754455235982503, 6.04053972326327524290034553283, 7.31294588560759777337170229601, 8.317336992464223774276434260188, 9.035264193972863157836162605958, 10.35345143304059655172538871084

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