Properties

Label 2-1152-16.3-c2-0-20
Degree $2$
Conductor $1152$
Sign $0.209 + 0.977i$
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  + (−2.41 + 2.41i)5-s − 11.8·7-s + (11.9 + 11.9i)11-s + (−2.08 − 2.08i)13-s − 23.1·17-s + (6.77 − 6.77i)19-s + 3.92·23-s + 13.3i·25-s + (−0.782 − 0.782i)29-s − 2.65i·31-s + (28.6 − 28.6i)35-s + (−37.2 + 37.2i)37-s − 69.1i·41-s + (29.2 + 29.2i)43-s − 68.0i·47-s + ⋯
L(s)  = 1  + (−0.482 + 0.482i)5-s − 1.69·7-s + (1.08 + 1.08i)11-s + (−0.160 − 0.160i)13-s − 1.36·17-s + (0.356 − 0.356i)19-s + 0.170·23-s + 0.534i·25-s + (−0.0269 − 0.0269i)29-s − 0.0855i·31-s + (0.818 − 0.818i)35-s + (−1.00 + 1.00i)37-s − 1.68i·41-s + (0.681 + 0.681i)43-s − 1.44i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.209 + 0.977i$
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: \((2,\ 1152,\ (\ :1),\ 0.209 + 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6357458168\)
\(L(\frac12)\) \(\approx\) \(0.6357458168\)
\(L(2)\) 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 \( 1 + (2.41 - 2.41i)T - 25iT^{2} \)
7 \( 1 + 11.8T + 49T^{2} \)
11 \( 1 + (-11.9 - 11.9i)T + 121iT^{2} \)
13 \( 1 + (2.08 + 2.08i)T + 169iT^{2} \)
17 \( 1 + 23.1T + 289T^{2} \)
19 \( 1 + (-6.77 + 6.77i)T - 361iT^{2} \)
23 \( 1 - 3.92T + 529T^{2} \)
29 \( 1 + (0.782 + 0.782i)T + 841iT^{2} \)
31 \( 1 + 2.65iT - 961T^{2} \)
37 \( 1 + (37.2 - 37.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 69.1iT - 1.68e3T^{2} \)
43 \( 1 + (-29.2 - 29.2i)T + 1.84e3iT^{2} \)
47 \( 1 + 68.0iT - 2.20e3T^{2} \)
53 \( 1 + (-40.3 + 40.3i)T - 2.80e3iT^{2} \)
59 \( 1 + (23.5 + 23.5i)T + 3.48e3iT^{2} \)
61 \( 1 + (65.9 + 65.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-40.9 + 40.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 98.1T + 5.04e3T^{2} \)
73 \( 1 + 74.2iT - 5.32e3T^{2} \)
79 \( 1 + 0.779iT - 6.24e3T^{2} \)
83 \( 1 + (-91.7 + 91.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 20.1iT - 7.92e3T^{2} \)
97 \( 1 + 86.6T + 9.40e3T^{2} \)
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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

−9.371981669182734333987369037402, −8.915575947354355010387363553506, −7.51765701706610827535176246435, −6.74201712644669986915407532467, −6.51002233938296503468710180208, −5.08420557330609871917710368879, −3.94300722804981929087798789417, −3.29063115327360264498801193710, −2.08949619943235972883398739258, −0.24093562520436377834727628781, 0.917612944159444951648195089162, 2.68041512950077635909157168359, 3.67123083201005112965563087170, 4.34872685347957748257136556768, 5.76868256984624504244765447358, 6.44046861672151428150372145925, 7.12292305416512201107025971944, 8.344430950796512111315521013708, 9.074775980260177130102170830963, 9.519043631395203359817985759985

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