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Properties

Label 2-1152-8.5-c3-0-22
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  + 10.4i·5-s + 6.42·7-s + 61.6i·11-s + 64.8i·13-s + 75.6·17-s − 10.3i·19-s + 156.·23-s + 16.3·25-s + 53.7i·29-s − 227.·31-s + 66.9i·35-s − 10.3i·37-s + 70.4·41-s − 298. i·43-s − 89.9·47-s + ⋯
L(s)  = 1  + 0.932i·5-s + 0.346·7-s + 1.69i·11-s + 1.38i·13-s + 1.07·17-s − 0.124i·19-s + 1.42·23-s + 0.131·25-s + 0.344i·29-s − 1.31·31-s + 0.323i·35-s − 0.0458i·37-s + 0.268·41-s − 1.05i·43-s − 0.279·47-s + ⋯

Functional equation

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

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

\(L(2)\) \(\approx\) \(2.042007123\)
\(L(\frac12)\) \(\approx\) \(2.042007123\)
\(L(\frac{5}{2})\) 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 \( 1 - 10.4iT - 125T^{2} \)
7 \( 1 - 6.42T + 343T^{2} \)
11 \( 1 - 61.6iT - 1.33e3T^{2} \)
13 \( 1 - 64.8iT - 2.19e3T^{2} \)
17 \( 1 - 75.6T + 4.91e3T^{2} \)
19 \( 1 + 10.3iT - 6.85e3T^{2} \)
23 \( 1 - 156.T + 1.21e4T^{2} \)
29 \( 1 - 53.7iT - 2.43e4T^{2} \)
31 \( 1 + 227.T + 2.97e4T^{2} \)
37 \( 1 + 10.3iT - 5.06e4T^{2} \)
41 \( 1 - 70.4T + 6.89e4T^{2} \)
43 \( 1 + 298. iT - 7.95e4T^{2} \)
47 \( 1 + 89.9T + 1.03e5T^{2} \)
53 \( 1 - 388. iT - 1.48e5T^{2} \)
59 \( 1 - 324iT - 2.05e5T^{2} \)
61 \( 1 - 324iT - 2.26e5T^{2} \)
67 \( 1 + 920. iT - 3.00e5T^{2} \)
71 \( 1 - 995.T + 3.57e5T^{2} \)
73 \( 1 - 362.T + 3.89e5T^{2} \)
79 \( 1 - 1.09e3T + 4.93e5T^{2} \)
83 \( 1 - 791. iT - 5.71e5T^{2} \)
89 \( 1 - 150.T + 7.04e5T^{2} \)
97 \( 1 + 1.87e3T + 9.12e5T^{2} \)
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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

−9.597888866655418868666742463102, −9.179738680337761933119659093763, −7.894268460857108261649912276522, −7.00578474548913727059353885026, −6.81367929900901273193759931657, −5.38731586690941028226723865844, −4.57062672916208044742490571136, −3.55003806008506280450855177536, −2.39615746850178358362119038403, −1.45949337423177490333232474641, 0.53958288353498521786914825917, 1.23075857334557121284452264106, 2.93800337824866416064038350755, 3.67641736853393534433766667374, 5.20283837247161520833944139718, 5.36574014281493879436250987525, 6.48557228123680577859011832239, 7.82851946648605993763262948830, 8.216926220159984072362683290865, 9.007250753273615396683056719886

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