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Properties

Label 2-1152-8.3-c2-0-3
Degree $2$
Conductor $1152$
Sign $-0.707 - 0.707i$
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  + 0.898i·5-s − 2.82i·7-s − 4.38·11-s − 13.7i·13-s − 17.5·17-s − 4.38·19-s + 22.0i·23-s + 24.1·25-s + 44.4i·29-s + 53.1i·31-s + 2.54·35-s − 35.1i·37-s − 37.5·41-s − 49.6·43-s − 38.4i·47-s + ⋯
L(s)  = 1  + 0.179i·5-s − 0.404i·7-s − 0.398·11-s − 1.06i·13-s − 1.03·17-s − 0.230·19-s + 0.958i·23-s + 0.967·25-s + 1.53i·29-s + 1.71i·31-s + 0.0726·35-s − 0.951i·37-s − 0.916·41-s − 1.15·43-s − 0.818i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5665470351\)
\(L(\frac12)\) \(\approx\) \(0.5665470351\)
\(L(2)\) 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 \( 1 - 0.898iT - 25T^{2} \)
7 \( 1 + 2.82iT - 49T^{2} \)
11 \( 1 + 4.38T + 121T^{2} \)
13 \( 1 + 13.7iT - 169T^{2} \)
17 \( 1 + 17.5T + 289T^{2} \)
19 \( 1 + 4.38T + 361T^{2} \)
23 \( 1 - 22.0iT - 529T^{2} \)
29 \( 1 - 44.4iT - 841T^{2} \)
31 \( 1 - 53.1iT - 961T^{2} \)
37 \( 1 + 35.1iT - 1.36e3T^{2} \)
41 \( 1 + 37.5T + 1.68e3T^{2} \)
43 \( 1 + 49.6T + 1.84e3T^{2} \)
47 \( 1 + 38.4iT - 2.20e3T^{2} \)
53 \( 1 + 1.70iT - 2.80e3T^{2} \)
59 \( 1 + 34.6T + 3.48e3T^{2} \)
61 \( 1 - 24.4iT - 3.72e3T^{2} \)
67 \( 1 + 93.7T + 4.48e3T^{2} \)
71 \( 1 - 123. iT - 5.04e3T^{2} \)
73 \( 1 + 10T + 5.32e3T^{2} \)
79 \( 1 - 131. iT - 6.24e3T^{2} \)
83 \( 1 + 110.T + 6.88e3T^{2} \)
89 \( 1 - 73.1T + 7.92e3T^{2} \)
97 \( 1 + 105.T + 9.40e3T^{2} \)
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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

−10.15980576998893365200432458310, −8.920689410726745002382201992026, −8.414186386137196520893278193984, −7.25171169047719682858569200968, −6.82036811720438231601806322153, −5.54692374776142321370270209232, −4.88568450338343393447401893601, −3.64118908673327097126789974055, −2.79130693577864188869297414258, −1.36625606762420919497736233700, 0.16678798527854457652783393846, 1.87865466543705960701325088973, 2.79433873926281439210583594860, 4.24820559361000098921891252219, 4.80133648499605798112827840113, 6.08715573576980810678184380612, 6.63665722325554178804090403915, 7.73624129402239228137083646647, 8.582341265208537766551693466846, 9.201147876030045622450889393942

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