Properties

Label 2-1152-72.59-c1-0-17
Degree $2$
Conductor $1152$
Sign $0.710 + 0.703i$
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  + (−1.65 + 0.524i)3-s + (−1.01 + 1.75i)5-s + (−4.09 + 2.36i)7-s + (2.44 − 1.73i)9-s + (−2.93 + 1.69i)11-s + (−5.51 − 3.18i)13-s + (0.751 − 3.42i)15-s − 4.87i·17-s + 1.48·19-s + (5.51 − 6.04i)21-s + (0.751 − 1.30i)23-s + (0.449 + 0.778i)25-s + (−3.13 + 4.14i)27-s + (3.49 + 6.04i)29-s + (5.93 + 3.42i)31-s + ⋯
L(s)  = 1  + (−0.953 + 0.302i)3-s + (−0.452 + 0.784i)5-s + (−1.54 + 0.893i)7-s + (0.816 − 0.577i)9-s + (−0.883 + 0.510i)11-s + (−1.53 − 0.883i)13-s + (0.193 − 0.884i)15-s − 1.18i·17-s + 0.340·19-s + (1.20 − 1.32i)21-s + (0.156 − 0.271i)23-s + (0.0898 + 0.155i)25-s + (−0.603 + 0.797i)27-s + (0.648 + 1.12i)29-s + (1.06 + 0.615i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2919032113\)
\(L(\frac12)\) \(\approx\) \(0.2919032113\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 - 0.524i)T \)
good5 \( 1 + (1.01 - 1.75i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (4.09 - 2.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.51 + 3.18i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.87iT - 17T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
23 \( 1 + (-0.751 + 1.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.49 - 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.93 - 3.42i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.86iT - 37T^{2} \)
41 \( 1 + (-4.62 - 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.76 - 4.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.09 + 7.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.96T + 53T^{2} \)
59 \( 1 + (7.88 + 4.55i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.51 + 3.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.48 + 9.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.68T + 71T^{2} \)
73 \( 1 - 0.449T + 73T^{2} \)
79 \( 1 + (5.93 - 3.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.06 + 3.50i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.142iT - 89T^{2} \)
97 \( 1 + (-0.724 - 1.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.900741389562299950894807189969, −9.238914123023602007968674882825, −7.79705839901928598603072094049, −6.99056305700415686436615562909, −6.47357771110597885500559669698, −5.31000699635562838190384298264, −4.82866765849398668166562754407, −3.11975911516867419166084217044, −2.79379894633875621441319675391, −0.21084567416037962076272138735, 0.78035815004963143893333396067, 2.57697219414131243507273408301, 4.03615742711324605361914285235, 4.64694776020012773559721034399, 5.79583474237896992450077065728, 6.51171569419134550493201549042, 7.37643287221596746008380512581, 8.013058449081410164650763625673, 9.285827532108444149737494246695, 10.04484859228185204693394078379

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