Properties

Label 2-1152-1.1-c3-0-41
Degree $2$
Conductor $1152$
Sign $-1$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 10·7-s + 4·11-s + 26·13-s − 14·17-s + 8·19-s + 148·23-s − 109·25-s − 72·29-s − 18·31-s + 40·35-s + 262·37-s + 378·41-s − 432·43-s + 148·47-s − 243·49-s − 360·53-s − 16·55-s + 428·59-s − 442·61-s − 104·65-s − 692·67-s + 540·71-s − 1.01e3·73-s − 40·77-s − 386·79-s − 108·83-s + ⋯
L(s)  = 1  − 0.357·5-s − 0.539·7-s + 0.109·11-s + 0.554·13-s − 0.199·17-s + 0.0965·19-s + 1.34·23-s − 0.871·25-s − 0.461·29-s − 0.104·31-s + 0.193·35-s + 1.16·37-s + 1.43·41-s − 1.53·43-s + 0.459·47-s − 0.708·49-s − 0.933·53-s − 0.0392·55-s + 0.944·59-s − 0.927·61-s − 0.198·65-s − 1.26·67-s + 0.902·71-s − 1.63·73-s − 0.0592·77-s − 0.549·79-s − 0.142·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1152} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4 T + p^{3} T^{2} \)
7 \( 1 + 10 T + p^{3} T^{2} \)
11 \( 1 - 4 T + p^{3} T^{2} \)
13 \( 1 - 2 p T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 - 8 T + p^{3} T^{2} \)
23 \( 1 - 148 T + p^{3} T^{2} \)
29 \( 1 + 72 T + p^{3} T^{2} \)
31 \( 1 + 18 T + p^{3} T^{2} \)
37 \( 1 - 262 T + p^{3} T^{2} \)
41 \( 1 - 378 T + p^{3} T^{2} \)
43 \( 1 + 432 T + p^{3} T^{2} \)
47 \( 1 - 148 T + p^{3} T^{2} \)
53 \( 1 + 360 T + p^{3} T^{2} \)
59 \( 1 - 428 T + p^{3} T^{2} \)
61 \( 1 + 442 T + p^{3} T^{2} \)
67 \( 1 + 692 T + p^{3} T^{2} \)
71 \( 1 - 540 T + p^{3} T^{2} \)
73 \( 1 + 1018 T + p^{3} T^{2} \)
79 \( 1 + 386 T + p^{3} T^{2} \)
83 \( 1 + 108 T + p^{3} T^{2} \)
89 \( 1 - 382 T + p^{3} T^{2} \)
97 \( 1 - 298 T + p^{3} T^{2} \)
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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

−9.113466060783897750477094990124, −8.169311962163469951875100147018, −7.36903489341410449781872074076, −6.49518704562489637171624912941, −5.70298783117378576200121784841, −4.58331263198406643137307536979, −3.66437452883277567842805175178, −2.74665138653021309039671269030, −1.31975510309162980575203880993, 0, 1.31975510309162980575203880993, 2.74665138653021309039671269030, 3.66437452883277567842805175178, 4.58331263198406643137307536979, 5.70298783117378576200121784841, 6.49518704562489637171624912941, 7.36903489341410449781872074076, 8.169311962163469951875100147018, 9.113466060783897750477094990124

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