Properties

Label 2-1152-8.5-c3-0-44
Degree $2$
Conductor $1152$
Sign $0.707 + 0.707i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22i·5-s − 92i·13-s + 104·17-s − 359·25-s − 130i·29-s − 396i·37-s − 472·41-s − 343·49-s − 518i·53-s − 468i·61-s + 2.02e3·65-s + 1.09e3·73-s + 2.28e3i·85-s − 176·89-s + 594·97-s + ⋯
L(s)  = 1  + 1.96i·5-s − 1.96i·13-s + 1.48·17-s − 2.87·25-s − 0.832i·29-s − 1.75i·37-s − 1.79·41-s − 49-s − 1.34i·53-s − 0.982i·61-s + 3.86·65-s + 1.76·73-s + 2.91i·85-s − 0.209·89-s + 0.621·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.496798537\)
\(L(\frac12)\) \(\approx\) \(1.496798537\)
\(L(\frac{5}{2})\) 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 \( 1 - 22iT - 125T^{2} \)
7 \( 1 + 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + 92iT - 2.19e3T^{2} \)
17 \( 1 - 104T + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 130iT - 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 396iT - 5.06e4T^{2} \)
41 \( 1 + 472T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 518iT - 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 + 468iT - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 176T + 7.04e5T^{2} \)
97 \( 1 - 594T + 9.12e5T^{2} \)
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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

−9.732953960746515101223919517471, −8.123635703419419196953461610066, −7.71420663486240630148559518068, −6.86029747532526183396515388647, −5.98153055837876300370539954559, −5.29994290890271469750969986721, −3.55210932264918301656539141648, −3.21580376133176961370314166557, −2.14246599443990054505614360189, −0.38698956432979135456994068457, 1.11774815787704034665666049057, 1.75864231969248505801397874591, 3.50755159231955498694552543596, 4.54563533469717558309558860576, 5.04436048771782344601069793970, 6.03161652659051841864626223642, 7.08276952125274223408447709788, 8.153446041973150164978631847715, 8.694770171897172745619544128182, 9.459575567983006801211560838401

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