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Properties

Label 2-1152-9.7-c1-0-25
Degree $2$
Conductor $1152$
Sign $0.884 - 0.466i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.762 + 1.55i)3-s + (0.705 + 1.22i)5-s + (1.17 − 2.02i)7-s + (−1.83 − 2.37i)9-s + (1.30 − 2.25i)11-s + (1.26 + 2.18i)13-s + (−2.43 + 0.165i)15-s + 4.94·17-s + 1.00·19-s + (2.26 + 3.36i)21-s + (−1.50 − 2.60i)23-s + (1.50 − 2.60i)25-s + (5.08 − 1.05i)27-s + (−0.0708 + 0.122i)29-s + (−4.77 − 8.26i)31-s + ⋯
L(s)  = 1  + (−0.440 + 0.897i)3-s + (0.315 + 0.546i)5-s + (0.442 − 0.766i)7-s + (−0.612 − 0.790i)9-s + (0.392 − 0.679i)11-s + (0.350 + 0.606i)13-s + (−0.629 + 0.0427i)15-s + 1.19·17-s + 0.231·19-s + (0.493 + 0.734i)21-s + (−0.314 − 0.544i)23-s + (0.300 − 0.521i)25-s + (0.979 − 0.202i)27-s + (−0.0131 + 0.0228i)29-s + (−0.856 − 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.635755181\)
\(L(\frac12)\) \(\approx\) \(1.635755181\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.762 - 1.55i)T \)
good5 \( 1 + (-0.705 - 1.22i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.17 + 2.02i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.30 + 2.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.26 - 2.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.94T + 17T^{2} \)
19 \( 1 - 1.00T + 19T^{2} \)
23 \( 1 + (1.50 + 2.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0708 - 0.122i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.77 + 8.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.00T + 37T^{2} \)
41 \( 1 + (-4.33 - 7.50i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.15 + 5.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.24 - 5.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.02T + 53T^{2} \)
59 \( 1 + (-5.64 - 9.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.45 - 5.99i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.154 + 0.268i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.24T + 71T^{2} \)
73 \( 1 + 6.78T + 73T^{2} \)
79 \( 1 + (4.99 - 8.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.47 - 6.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + (-7.44 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.900209381585410809445636911234, −9.304927422740215997169066518717, −8.261448997118398880005471375091, −7.37170787711403356293204845394, −6.23559355955967712693417558612, −5.78828163382819729204665426626, −4.49524159721015294784357152450, −3.88010309172953264633220023923, −2.77911347462985339400941677536, −0.989769989168480871328111870616, 1.13468720726333532403441617160, 2.03527863658142283191770112420, 3.37341071456529645510869600100, 4.97010355658943709524620522516, 5.46926978161993148178188681511, 6.26355325310889949766912308806, 7.36574939175786519700096650930, 7.977285399981125173854127503642, 8.861695304098726656588478068377, 9.616798388388190806047472163658

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