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Properties

Label 2-1152-48.29-c2-0-2
Degree $2$
Conductor $1152$
Sign $0.373 - 0.927i$
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  + (−3.90 − 3.90i)5-s − 0.778i·7-s + (−4.29 − 4.29i)11-s + (−6.44 − 6.44i)13-s + 31.3i·17-s + (−1.11 − 1.11i)19-s − 34.2·23-s + 5.47i·25-s + (−8.77 + 8.77i)29-s + 50.8·31-s + (−3.03 + 3.03i)35-s + (29.3 − 29.3i)37-s + 31.4·41-s + (−55.9 + 55.9i)43-s − 26.5i·47-s + ⋯
L(s)  = 1  + (−0.780 − 0.780i)5-s − 0.111i·7-s + (−0.390 − 0.390i)11-s + (−0.495 − 0.495i)13-s + 1.84i·17-s + (−0.0588 − 0.0588i)19-s − 1.48·23-s + 0.218i·25-s + (−0.302 + 0.302i)29-s + 1.64·31-s + (−0.0868 + 0.0868i)35-s + (0.792 − 0.792i)37-s + 0.766·41-s + (−1.30 + 1.30i)43-s − 0.564i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.373 - 0.927i$
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: \((2,\ 1152,\ (\ :1),\ 0.373 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8411374460\)
\(L(\frac12)\) \(\approx\) \(0.8411374460\)
\(L(2)\) 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 \( 1 + (3.90 + 3.90i)T + 25iT^{2} \)
7 \( 1 + 0.778iT - 49T^{2} \)
11 \( 1 + (4.29 + 4.29i)T + 121iT^{2} \)
13 \( 1 + (6.44 + 6.44i)T + 169iT^{2} \)
17 \( 1 - 31.3iT - 289T^{2} \)
19 \( 1 + (1.11 + 1.11i)T + 361iT^{2} \)
23 \( 1 + 34.2T + 529T^{2} \)
29 \( 1 + (8.77 - 8.77i)T - 841iT^{2} \)
31 \( 1 - 50.8T + 961T^{2} \)
37 \( 1 + (-29.3 + 29.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 31.4T + 1.68e3T^{2} \)
43 \( 1 + (55.9 - 55.9i)T - 1.84e3iT^{2} \)
47 \( 1 + 26.5iT - 2.20e3T^{2} \)
53 \( 1 + (9.76 + 9.76i)T + 2.80e3iT^{2} \)
59 \( 1 + (-54.3 - 54.3i)T + 3.48e3iT^{2} \)
61 \( 1 + (-47.1 - 47.1i)T + 3.72e3iT^{2} \)
67 \( 1 + (-66.1 - 66.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 75.9T + 5.04e3T^{2} \)
73 \( 1 - 24.1iT - 5.32e3T^{2} \)
79 \( 1 + 80.5T + 6.24e3T^{2} \)
83 \( 1 + (82.6 - 82.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 82.6T + 7.92e3T^{2} \)
97 \( 1 - 48.9T + 9.40e3T^{2} \)
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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

−9.937083706687656849477935107111, −8.598894987779066671282639989726, −8.243300731154730314674336743830, −7.54722589269456989063860083300, −6.29405861279546984887223533341, −5.53580863614383353733838601008, −4.40432700441040780399348185802, −3.81253851070719781270888535798, −2.44858964455503266986864291169, −0.976767690075579381466174671070, 0.30477667347074278073682813799, 2.22143848907212976579832413023, 3.08689239349436923951584040348, 4.22217706184884761686255669097, 5.02617101276890791765932505450, 6.23196739070789680833384981009, 7.12633106454933490345958697699, 7.63903514870179530170735542820, 8.515121276492998490976343459427, 9.708034709443235910426478007171

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