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Properties

Label 2-1152-16.3-c2-0-22
Degree $2$
Conductor $1152$
Sign $0.895 + 0.444i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.227 − 0.227i)5-s + 3.90·7-s + (−2.21 − 2.21i)11-s + (−5.08 − 5.08i)13-s + 18.8·17-s + (−11.7 + 11.7i)19-s + 35.4·23-s + 24.8i·25-s + (−21.2 − 21.2i)29-s − 35.9i·31-s + (0.888 − 0.888i)35-s + (34.4 − 34.4i)37-s − 44.1i·41-s + (28.3 + 28.3i)43-s + 32.8i·47-s + ⋯
L(s)  = 1  + (0.0455 − 0.0455i)5-s + 0.557·7-s + (−0.200 − 0.200i)11-s + (−0.391 − 0.391i)13-s + 1.10·17-s + (−0.619 + 0.619i)19-s + 1.54·23-s + 0.995i·25-s + (−0.732 − 0.732i)29-s − 1.16i·31-s + (0.0253 − 0.0253i)35-s + (0.930 − 0.930i)37-s − 1.07i·41-s + (0.658 + 0.658i)43-s + 0.699i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.054784173\)
\(L(\frac12)\) \(\approx\) \(2.054784173\)
\(L(2)\) 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 \( 1 + (-0.227 + 0.227i)T - 25iT^{2} \)
7 \( 1 - 3.90T + 49T^{2} \)
11 \( 1 + (2.21 + 2.21i)T + 121iT^{2} \)
13 \( 1 + (5.08 + 5.08i)T + 169iT^{2} \)
17 \( 1 - 18.8T + 289T^{2} \)
19 \( 1 + (11.7 - 11.7i)T - 361iT^{2} \)
23 \( 1 - 35.4T + 529T^{2} \)
29 \( 1 + (21.2 + 21.2i)T + 841iT^{2} \)
31 \( 1 + 35.9iT - 961T^{2} \)
37 \( 1 + (-34.4 + 34.4i)T - 1.36e3iT^{2} \)
41 \( 1 + 44.1iT - 1.68e3T^{2} \)
43 \( 1 + (-28.3 - 28.3i)T + 1.84e3iT^{2} \)
47 \( 1 - 32.8iT - 2.20e3T^{2} \)
53 \( 1 + (-42.1 + 42.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-66.9 - 66.9i)T + 3.48e3iT^{2} \)
61 \( 1 + (17.3 + 17.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (39.6 - 39.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 63.0T + 5.04e3T^{2} \)
73 \( 1 - 75.4iT - 5.32e3T^{2} \)
79 \( 1 + 59.1iT - 6.24e3T^{2} \)
83 \( 1 + (-71.2 + 71.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 150. iT - 7.92e3T^{2} \)
97 \( 1 - 51.5T + 9.40e3T^{2} \)
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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

−9.546980890712287082288371693641, −8.741393322911655796987720600263, −7.74388029832737340480451010016, −7.35370362708317304646952846211, −5.95774619354952003688293262315, −5.39688549691605113003311417819, −4.33946071588047215263057037513, −3.28693382113487083142844810068, −2.11822659020133503621766070141, −0.78111900131930658359407088467, 1.01757265630884592922856472595, 2.31215574398312601074267925440, 3.38597489523762735349454749265, 4.67633859561191252931373942184, 5.19499869981817061995638141350, 6.42442925426174262069638803644, 7.18514342440622113394638661496, 8.035267208245581961884007593457, 8.839930607705374704298867331040, 9.648497200394975437658404450980

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