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Properties

Label 2-1152-48.35-c3-0-17
Degree $2$
Conductor $1152$
Sign $0.682 + 0.731i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.7 − 11.7i)5-s − 11.9·7-s + (−36.9 + 36.9i)11-s + (−20.4 − 20.4i)13-s + 81.1i·17-s + (−29.9 + 29.9i)19-s + 163. i·23-s + 151. i·25-s + (201. − 201. i)29-s + 43.1i·31-s + (141. + 141. i)35-s + (−100. + 100. i)37-s + 345.·41-s + (−326. − 326. i)43-s − 116.·47-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)5-s − 0.647·7-s + (−1.01 + 1.01i)11-s + (−0.436 − 0.436i)13-s + 1.15i·17-s + (−0.361 + 0.361i)19-s + 1.48i·23-s + 1.21i·25-s + (1.28 − 1.28i)29-s + 0.249i·31-s + (0.681 + 0.681i)35-s + (−0.444 + 0.444i)37-s + 1.31·41-s + (−1.15 − 1.15i)43-s − 0.361·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.731i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.682 + 0.731i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.682 + 0.731i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7150816906\)
\(L(\frac12)\) \(\approx\) \(0.7150816906\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.7 + 11.7i)T + 125iT^{2} \)
7 \( 1 + 11.9T + 343T^{2} \)
11 \( 1 + (36.9 - 36.9i)T - 1.33e3iT^{2} \)
13 \( 1 + (20.4 + 20.4i)T + 2.19e3iT^{2} \)
17 \( 1 - 81.1iT - 4.91e3T^{2} \)
19 \( 1 + (29.9 - 29.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 163. iT - 1.21e4T^{2} \)
29 \( 1 + (-201. + 201. i)T - 2.43e4iT^{2} \)
31 \( 1 - 43.1iT - 2.97e4T^{2} \)
37 \( 1 + (100. - 100. i)T - 5.06e4iT^{2} \)
41 \( 1 - 345.T + 6.89e4T^{2} \)
43 \( 1 + (326. + 326. i)T + 7.95e4iT^{2} \)
47 \( 1 + 116.T + 1.03e5T^{2} \)
53 \( 1 + (-16.3 - 16.3i)T + 1.48e5iT^{2} \)
59 \( 1 + (46.4 - 46.4i)T - 2.05e5iT^{2} \)
61 \( 1 + (69.6 + 69.6i)T + 2.26e5iT^{2} \)
67 \( 1 + (157. - 157. i)T - 3.00e5iT^{2} \)
71 \( 1 + 690. iT - 3.57e5T^{2} \)
73 \( 1 + 799. iT - 3.89e5T^{2} \)
79 \( 1 + 763. iT - 4.93e5T^{2} \)
83 \( 1 + (-940. - 940. i)T + 5.71e5iT^{2} \)
89 \( 1 + 660.T + 7.04e5T^{2} \)
97 \( 1 + 821.T + 9.12e5T^{2} \)
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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

−9.369174753962257257037905917449, −8.173376870039162895215702909612, −7.972766851112069027058020239331, −6.96743750435291257861800059041, −5.81643503780776789524758210546, −4.89389309968174502761512510517, −4.16239682494043653468113409718, −3.16217739543951973298054694466, −1.78517924750548536206073823414, −0.34270986320875583778829614436, 0.51465235834974732371242450753, 2.74843152330732326145643203742, 2.99052576889532772766845401311, 4.23288408678480595760738390204, 5.19945808853445728884173301196, 6.51707607664168495353790958014, 6.91060764799133547484119611040, 7.87135765385748302431998310319, 8.562307563446982827386715030292, 9.592430664843987891332947815377

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