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Properties

Label 2-891-99.68-c1-0-42
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: $\chi_{891} (134, \cdot )$
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

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

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