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Properties

Label 2-1152-48.29-c2-0-30
Degree $2$
Conductor $1152$
Sign $-0.696 - 0.717i$
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  + (−5.52 − 5.52i)5-s − 7.79i·7-s + (2.48 + 2.48i)11-s + (−13.3 − 13.3i)13-s − 5.86i·17-s + (−18.5 − 18.5i)19-s + 34.3·23-s + 36.0i·25-s + (21.4 − 21.4i)29-s − 30.6·31-s + (−43.0 + 43.0i)35-s + (−30.3 + 30.3i)37-s − 3.12·41-s + (9.94 − 9.94i)43-s − 38.4i·47-s + ⋯
L(s)  = 1  + (−1.10 − 1.10i)5-s − 1.11i·7-s + (0.225 + 0.225i)11-s + (−1.02 − 1.02i)13-s − 0.344i·17-s + (−0.977 − 0.977i)19-s + 1.49·23-s + 1.44i·25-s + (0.740 − 0.740i)29-s − 0.987·31-s + (−1.23 + 1.23i)35-s + (−0.819 + 0.819i)37-s − 0.0762·41-s + (0.231 − 0.231i)43-s − 0.817i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5298982419\)
\(L(\frac12)\) \(\approx\) \(0.5298982419\)
\(L(2)\) 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 \( 1 + (5.52 + 5.52i)T + 25iT^{2} \)
7 \( 1 + 7.79iT - 49T^{2} \)
11 \( 1 + (-2.48 - 2.48i)T + 121iT^{2} \)
13 \( 1 + (13.3 + 13.3i)T + 169iT^{2} \)
17 \( 1 + 5.86iT - 289T^{2} \)
19 \( 1 + (18.5 + 18.5i)T + 361iT^{2} \)
23 \( 1 - 34.3T + 529T^{2} \)
29 \( 1 + (-21.4 + 21.4i)T - 841iT^{2} \)
31 \( 1 + 30.6T + 961T^{2} \)
37 \( 1 + (30.3 - 30.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 3.12T + 1.68e3T^{2} \)
43 \( 1 + (-9.94 + 9.94i)T - 1.84e3iT^{2} \)
47 \( 1 + 38.4iT - 2.20e3T^{2} \)
53 \( 1 + (-61.1 - 61.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (-2.98 - 2.98i)T + 3.48e3iT^{2} \)
61 \( 1 + (3.88 + 3.88i)T + 3.72e3iT^{2} \)
67 \( 1 + (-47.0 - 47.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 97.5T + 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + 96.8T + 6.24e3T^{2} \)
83 \( 1 + (88.8 - 88.8i)T - 6.88e3iT^{2} \)
89 \( 1 - 54.8T + 7.92e3T^{2} \)
97 \( 1 + 5.00T + 9.40e3T^{2} \)
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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

−8.900483313751026634710576762678, −8.336899224395920568536202672555, −7.29044257421725986838232773061, −7.03956786676071761748031787628, −5.37314783182496092249018194078, −4.63216383044000374387794415886, −4.01393858637612795602783182316, −2.79510367961902335468921065723, −1.00058941197342458830593057526, −0.19075466799703773321779744708, 1.98610777944064575495225312196, 2.99592585790992365593182445788, 3.88987470330407975010283327190, 4.92943351531641064319816281227, 6.06769096540261012144374524982, 6.91429154918884446117462670261, 7.49607229710994491851006238728, 8.594983411026816743515749291170, 9.093524200195235537392717903510, 10.27436353810581307089188660204

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