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Properties

Label 2-1152-96.83-c1-0-10
Degree $2$
Conductor $1152$
Sign $0.663 + 0.748i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
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  + (0.366 + 0.885i)5-s + (−1.21 − 1.21i)7-s + (−0.545 − 1.31i)11-s + (0.0270 + 0.0112i)13-s + 1.32·17-s + (1.73 − 4.19i)19-s + (−0.934 − 0.934i)23-s + (2.88 − 2.88i)25-s + (9.35 + 3.87i)29-s − 9.74i·31-s + (0.630 − 1.52i)35-s + (−6.28 + 2.60i)37-s + (3.42 − 3.42i)41-s + (−0.997 + 0.413i)43-s + 6.21i·47-s + ⋯
L(s)  = 1  + (0.164 + 0.396i)5-s + (−0.459 − 0.459i)7-s + (−0.164 − 0.396i)11-s + (0.00750 + 0.00310i)13-s + 0.321·17-s + (0.398 − 0.961i)19-s + (−0.194 − 0.194i)23-s + (0.577 − 0.577i)25-s + (1.73 + 0.719i)29-s − 1.75i·31-s + (0.106 − 0.257i)35-s + (−1.03 + 0.427i)37-s + (0.534 − 0.534i)41-s + (−0.152 + 0.0630i)43-s + 0.906i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.663 + 0.748i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.663 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.486575335\)
\(L(\frac12)\) \(\approx\) \(1.486575335\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.366 - 0.885i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.21 + 1.21i)T + 7iT^{2} \)
11 \( 1 + (0.545 + 1.31i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.0270 - 0.0112i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.32T + 17T^{2} \)
19 \( 1 + (-1.73 + 4.19i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.934 + 0.934i)T + 23iT^{2} \)
29 \( 1 + (-9.35 - 3.87i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 9.74iT - 31T^{2} \)
37 \( 1 + (6.28 - 2.60i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.42 + 3.42i)T - 41iT^{2} \)
43 \( 1 + (0.997 - 0.413i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 6.21iT - 47T^{2} \)
53 \( 1 + (-2.94 + 1.22i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-10.4 + 4.32i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.76 + 6.68i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-10.1 - 4.18i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (7.38 - 7.38i)T - 71iT^{2} \)
73 \( 1 + (8.30 + 8.30i)T + 73iT^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 + (-11.4 - 4.74i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (7.93 + 7.93i)T + 89iT^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−9.855729257841450197145108817485, −8.860968886685564965233514817413, −8.074693948677379125726394396339, −7.03575856316534873991091173911, −6.50136522884165582800719721951, −5.46866878251426490252685033337, −4.45511521181177639709930968315, −3.34858600589336920596485786495, −2.47396114421965305020481035472, −0.72666569274124312226415731508, 1.29003274361309798627900820166, 2.64982528385408421100056703119, 3.68139094010987866700539656316, 4.88640366624430635789370236556, 5.60527306497156132975456641801, 6.54957900422221268166563893426, 7.42106961828353325111631343093, 8.427778546032003741338497220731, 9.016107102751701996376304832544, 10.02294218121372240744988729588

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