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Properties

Label 2-1150-5.4-c3-0-1
Degree $2$
Conductor $1150$
Sign $0.447 - 0.894i$
Analytic cond. $67.8521$
Root an. cond. $8.23724$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 10.1i·3-s − 4·4-s + 20.2·6-s − 24.6i·7-s − 8i·8-s − 75.5·9-s − 17.3·11-s + 40.5i·12-s − 4.00i·13-s + 49.2·14-s + 16·16-s + 48.2i·17-s − 151. i·18-s − 79.3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.94i·3-s − 0.5·4-s + 1.37·6-s − 1.33i·7-s − 0.353i·8-s − 2.79·9-s − 0.476·11-s + 0.974i·12-s − 0.0854i·13-s + 0.940·14-s + 0.250·16-s + 0.688i·17-s − 1.97i·18-s − 0.957·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(67.8521\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1921102874\)
\(L(\frac12)\) \(\approx\) \(0.1921102874\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 \)
23 \( 1 - 23iT \)
good3 \( 1 + 10.1iT - 27T^{2} \)
7 \( 1 + 24.6iT - 343T^{2} \)
11 \( 1 + 17.3T + 1.33e3T^{2} \)
13 \( 1 + 4.00iT - 2.19e3T^{2} \)
17 \( 1 - 48.2iT - 4.91e3T^{2} \)
19 \( 1 + 79.3T + 6.85e3T^{2} \)
29 \( 1 - 254.T + 2.43e4T^{2} \)
31 \( 1 + 220.T + 2.97e4T^{2} \)
37 \( 1 + 422. iT - 5.06e4T^{2} \)
41 \( 1 + 170.T + 6.89e4T^{2} \)
43 \( 1 - 228. iT - 7.95e4T^{2} \)
47 \( 1 - 580. iT - 1.03e5T^{2} \)
53 \( 1 - 260. iT - 1.48e5T^{2} \)
59 \( 1 + 353.T + 2.05e5T^{2} \)
61 \( 1 + 80.6T + 2.26e5T^{2} \)
67 \( 1 + 820. iT - 3.00e5T^{2} \)
71 \( 1 - 614.T + 3.57e5T^{2} \)
73 \( 1 + 511. iT - 3.89e5T^{2} \)
79 \( 1 + 160.T + 4.93e5T^{2} \)
83 \( 1 - 32.5iT - 5.71e5T^{2} \)
89 \( 1 - 25.0T + 7.04e5T^{2} \)
97 \( 1 + 249. iT - 9.12e5T^{2} \)
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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

−9.216449971680407050017996191417, −8.272768498564680459012897514655, −7.73786016585773788984918676524, −7.15669307280296858392198757587, −6.42103289589688050041964026473, −5.78761560710125910177857116706, −4.52158078583373452408434339192, −3.24643849852143620491621538596, −1.92895323824161830021227030763, −0.878206225962369372861895647440, 0.05613313390177717212234510882, 2.32341866345266707873891956109, 3.00911281892576539223792016648, 3.96861167659493540577962425607, 5.00534256251555779231360678314, 5.30738968611444071522452618295, 6.44366018912813131174526223851, 8.341440865638600252971662694745, 8.682808998156499167900641994455, 9.413392164320556991058220245003

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