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Properties

Label 2-1152-9.4-c1-0-4
Degree $2$
Conductor $1152$
Sign $0.107 - 0.994i$
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  + (−1.71 − 0.243i)3-s + (−1.34 + 2.32i)5-s + (−2.48 − 4.30i)7-s + (2.88 + 0.835i)9-s + (1.26 + 2.19i)11-s + (2.21 − 3.83i)13-s + (2.86 − 3.65i)15-s − 2.43·17-s − 4.18·19-s + (3.21 + 7.99i)21-s + (0.570 − 0.988i)23-s + (−1.09 − 1.89i)25-s + (−4.73 − 2.13i)27-s + (3.00 + 5.20i)29-s + (2.65 − 4.59i)31-s + ⋯
L(s)  = 1  + (−0.990 − 0.140i)3-s + (−0.599 + 1.03i)5-s + (−0.939 − 1.62i)7-s + (0.960 + 0.278i)9-s + (0.382 + 0.662i)11-s + (0.614 − 1.06i)13-s + (0.739 − 0.943i)15-s − 0.590·17-s − 0.959·19-s + (0.701 + 1.74i)21-s + (0.118 − 0.206i)23-s + (−0.218 − 0.378i)25-s + (−0.911 − 0.410i)27-s + (0.557 + 0.966i)29-s + (0.476 − 0.825i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.107 - 0.994i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.107 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5539270733\)
\(L(\frac12)\) \(\approx\) \(0.5539270733\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.243i)T \)
good5 \( 1 + (1.34 - 2.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.48 + 4.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.26 - 2.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.21 + 3.83i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.43T + 17T^{2} \)
19 \( 1 + 4.18T + 19T^{2} \)
23 \( 1 + (-0.570 + 0.988i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.00 - 5.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.65 + 4.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.241T + 37T^{2} \)
41 \( 1 + (3.21 - 5.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.57 - 9.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.37 - 4.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.38T + 53T^{2} \)
59 \( 1 + (5.40 - 9.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.16 - 7.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.13 + 1.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.52T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + (-3.14 - 5.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.738 - 1.27i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + (-5.89 - 10.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.33002245174255351829471819893, −9.500125251566003036808441172257, −7.974642242715641021216287921985, −7.30484007787686714836177281957, −6.59357644268829697827276559729, −6.21028946916420723083783169477, −4.60642998810417889441441835450, −3.96429569285183043186388583823, −2.96107382545968950900657665927, −1.02342626039179182839643292108, 0.34269089771804866977420507089, 1.99583091326405673392510162343, 3.59826937048986691582818209127, 4.49997783216952416975514165551, 5.38594531031539389449014338935, 6.27653863975441813343022813838, 6.67097102850351513315504801837, 8.272409572171982722281087965254, 8.923920651702120000723354422677, 9.333410154580688960938394172516

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