Properties

Label 2-1152-24.5-c0-0-2
Degree $2$
Conductor $1152$
Sign $0.169 + 0.985i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·5-s − 2i·13-s − 1.41i·17-s + 1.00·25-s + 1.41·29-s − 1.41i·41-s − 49-s − 1.41·53-s + 2.82i·65-s + 2.00i·85-s + 1.41i·89-s + 1.41·101-s + 2i·109-s − 1.41i·113-s + ⋯
L(s)  = 1  − 1.41·5-s − 2i·13-s − 1.41i·17-s + 1.00·25-s + 1.41·29-s − 1.41i·41-s − 49-s − 1.41·53-s + 2.82i·65-s + 2.00i·85-s + 1.41i·89-s + 1.41·101-s + 2i·109-s − 1.41i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :0),\ 0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6888992838\)
\(L(\frac12)\) \(\approx\) \(0.6888992838\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.913347031350007168799638457063, −8.835816872440099270693228017443, −7.982791495195756971958434785624, −7.59676607051189320959015277668, −6.62493620497504100776350074793, −5.39229759305115307216307119454, −4.65305696879996544277111566046, −3.49831560605953047502453314750, −2.80269654150583840512696627740, −0.64291267251029234845886976374, 1.66556699678536343319573575223, 3.21110176112132681611861235441, 4.19946092318241915619863649394, 4.65905070814679167101539162825, 6.24464345949396607014312935750, 6.82479710757392465753180640923, 7.85248963114647365968400683649, 8.417535917422872233665348684111, 9.245880145937487278194261597930, 10.22216496011576365503574831360

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