Properties

Label 2-1152-72.59-c1-0-21
Degree $2$
Conductor $1152$
Sign $0.946 - 0.322i$
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  + (1.54 − 0.783i)3-s + (−1.07 + 1.85i)5-s + (−1.13 + 0.652i)7-s + (1.77 − 2.42i)9-s + (2.36 − 1.36i)11-s + (1.16 + 0.673i)13-s + (−0.201 + 3.70i)15-s + 4.27i·17-s + 2.16·19-s + (−1.23 + 1.89i)21-s + (1.01 − 1.76i)23-s + (0.205 + 0.355i)25-s + (0.841 − 5.12i)27-s + (2.72 + 4.72i)29-s + (6.09 + 3.51i)31-s + ⋯
L(s)  = 1  + (0.891 − 0.452i)3-s + (−0.479 + 0.829i)5-s + (−0.427 + 0.246i)7-s + (0.590 − 0.806i)9-s + (0.711 − 0.411i)11-s + (0.323 + 0.186i)13-s + (−0.0519 + 0.956i)15-s + 1.03i·17-s + 0.495·19-s + (−0.269 + 0.413i)21-s + (0.212 − 0.368i)23-s + (0.0410 + 0.0711i)25-s + (0.161 − 0.986i)27-s + (0.506 + 0.877i)29-s + (1.09 + 0.631i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.946 - 0.322i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.946 - 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.115470115\)
\(L(\frac12)\) \(\approx\) \(2.115470115\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 + 0.783i)T \)
good5 \( 1 + (1.07 - 1.85i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.13 - 0.652i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.36 + 1.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.16 - 0.673i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.27iT - 17T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 + (-1.01 + 1.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.72 - 4.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.09 - 3.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 + (-0.596 - 0.344i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.22 - 2.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.56 - 11.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.56T + 53T^{2} \)
59 \( 1 + (-8.97 - 5.18i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.68 + 4.43i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.23 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 + (0.680 - 0.393i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.82 - 1.05i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 + (8.05 + 13.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.670966880705650511340029222994, −8.921832868513775370088148105603, −8.250965399683185803337775605431, −7.33036376339366935923856013857, −6.64386569022830568827142489403, −5.95096410068490102825245345989, −4.29761452451612601845636002052, −3.42625165226559878208374352675, −2.76088626167107958037445969739, −1.31373371097801847486522894323, 1.00593520707197934707910524430, 2.57484354521991528000069683018, 3.64600338167904078736729071782, 4.41756915355642716858262384033, 5.19400014181012790038701447396, 6.58726322511475171569759157944, 7.43237431277187056118581875831, 8.314338670379596379497240272482, 8.857504319156488527830911574403, 9.813137257446085487148395272171

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