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Properties

Label 2-1152-16.11-c2-0-8
Degree $2$
Conductor $1152$
Sign $0.475 - 0.879i$
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  + (−4.62 − 4.62i)5-s + 3.04·7-s + (−9.15 + 9.15i)11-s + (5.78 − 5.78i)13-s − 17.6·17-s + (1.15 + 1.15i)19-s + 3.45·23-s + 17.8i·25-s + (12.1 − 12.1i)29-s + 38.5i·31-s + (−14.1 − 14.1i)35-s + (0.0972 + 0.0972i)37-s + 51.5i·41-s + (1.70 − 1.70i)43-s − 24.1i·47-s + ⋯
L(s)  = 1  + (−0.925 − 0.925i)5-s + 0.435·7-s + (−0.831 + 0.831i)11-s + (0.444 − 0.444i)13-s − 1.03·17-s + (0.0606 + 0.0606i)19-s + 0.150·23-s + 0.712i·25-s + (0.420 − 0.420i)29-s + 1.24i·31-s + (−0.403 − 0.403i)35-s + (0.00262 + 0.00262i)37-s + 1.25i·41-s + (0.0395 − 0.0395i)43-s − 0.513i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.475 - 0.879i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.475 - 0.879i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.009958349\)
\(L(\frac12)\) \(\approx\) \(1.009958349\)
\(L(2)\) 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 \( 1 + (4.62 + 4.62i)T + 25iT^{2} \)
7 \( 1 - 3.04T + 49T^{2} \)
11 \( 1 + (9.15 - 9.15i)T - 121iT^{2} \)
13 \( 1 + (-5.78 + 5.78i)T - 169iT^{2} \)
17 \( 1 + 17.6T + 289T^{2} \)
19 \( 1 + (-1.15 - 1.15i)T + 361iT^{2} \)
23 \( 1 - 3.45T + 529T^{2} \)
29 \( 1 + (-12.1 + 12.1i)T - 841iT^{2} \)
31 \( 1 - 38.5iT - 961T^{2} \)
37 \( 1 + (-0.0972 - 0.0972i)T + 1.36e3iT^{2} \)
41 \( 1 - 51.5iT - 1.68e3T^{2} \)
43 \( 1 + (-1.70 + 1.70i)T - 1.84e3iT^{2} \)
47 \( 1 + 24.1iT - 2.20e3T^{2} \)
53 \( 1 + (-27.0 - 27.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (-19.5 + 19.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (16.7 - 16.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-75.8 - 75.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 134.T + 5.04e3T^{2} \)
73 \( 1 - 112. iT - 5.32e3T^{2} \)
79 \( 1 + 135. iT - 6.24e3T^{2} \)
83 \( 1 + (-74.9 - 74.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 31.4iT - 7.92e3T^{2} \)
97 \( 1 - 31.5T + 9.40e3T^{2} \)
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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

−9.724865772975707699362588276817, −8.659884266093022666151045381270, −8.220494059674924703124156566286, −7.45920916302465408236508039481, −6.48223820780125026814592998715, −5.14163586622919933671770061010, −4.70298628808896434415955930990, −3.72798114466357988032424353460, −2.37888539854800271455791720371, −0.995725447331528654900335356291, 0.36582841150529741602320954291, 2.17895952697658352631442953173, 3.24346188490794340159391436261, 4.08577818189622806826178611343, 5.13464615929505603251419125248, 6.24654063281099088129885281486, 7.01290428845593125485332158201, 7.87433788043988228808860109466, 8.454922603463098899806051458488, 9.410959417061366609816513580772

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