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Properties

Label 2-1152-72.61-c1-0-20
Degree $2$
Conductor $1152$
Sign $-0.173 + 0.984i$
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  + (−1.22 − 1.22i)3-s + (−2.03 + 1.17i)5-s + (−2.27 + 3.93i)7-s + 2.99i·9-s + (−0.448 − 0.258i)11-s + (−3.52 + 2.03i)13-s + (3.93 + 1.05i)15-s + 4.73·17-s − 4.89i·19-s + (7.60 − 2.03i)21-s + (−3.93 − 6.81i)23-s + (0.267 − 0.464i)25-s + (3.67 − 3.67i)27-s + (3.52 + 2.03i)29-s + (−0.608 − 1.05i)31-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.911 + 0.526i)5-s + (−0.858 + 1.48i)7-s + 0.999i·9-s + (−0.135 − 0.0780i)11-s + (−0.978 + 0.565i)13-s + (1.01 + 0.272i)15-s + 1.14·17-s − 1.12i·19-s + (1.65 − 0.444i)21-s + (−0.820 − 1.42i)23-s + (0.0535 − 0.0928i)25-s + (0.707 − 0.707i)27-s + (0.655 + 0.378i)29-s + (−0.109 − 0.189i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3356536619\)
\(L(\frac12)\) \(\approx\) \(0.3356536619\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (2.03 - 1.17i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.27 - 3.93i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.448 + 0.258i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.52 - 2.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 + 4.89iT - 19T^{2} \)
23 \( 1 + (3.93 + 6.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.52 - 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.608 + 1.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.1iT - 37T^{2} \)
41 \( 1 + (-3.86 - 6.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.46 + 2.00i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.05 - 1.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.42iT - 53T^{2} \)
59 \( 1 + (-0.120 + 0.0693i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.52 - 2.03i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.208 - 0.120i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.98T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + (-3.93 + 6.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.81 + 3.93i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.19T + 89T^{2} \)
97 \( 1 + (-9.33 + 16.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.576035962155835920886457846760, −8.677460716284639839438821014594, −7.74643603340884486210974598110, −7.03525406984863423418448477551, −6.26531243018666708526764748120, −5.49189428852357325530576317554, −4.47769890083510183425725641761, −3.01407459435287599535638388820, −2.28434724036373659904161178309, −0.20630763014686587918774294146, 0.947435906389955279223890898882, 3.38495348666352757105956462899, 3.84539198906623508905293516763, 4.78005126981853100173224837830, 5.66891108400500066357654128193, 6.71813494777430825289283126854, 7.62651352603401478317820982680, 8.164560531343968963457388009960, 9.753455324866215836991006268066, 9.917028128557573398839865471025

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