Properties

Label 2-1152-1.1-c1-0-5
Degree $2$
Conductor $1152$
Sign $1$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 4·11-s + 6·13-s − 6·17-s − 4·23-s − 5·25-s − 4·29-s + 10·31-s + 2·37-s + 2·41-s + 8·43-s + 12·47-s − 3·49-s + 12·53-s + 4·59-s + 2·61-s + 4·67-s + 4·71-s − 10·73-s + 8·77-s − 6·79-s − 12·83-s − 2·89-s + 12·91-s − 6·97-s − 4·101-s − 10·103-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.834·23-s − 25-s − 0.742·29-s + 1.79·31-s + 0.328·37-s + 0.312·41-s + 1.21·43-s + 1.75·47-s − 3/7·49-s + 1.64·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.474·71-s − 1.17·73-s + 0.911·77-s − 0.675·79-s − 1.31·83-s − 0.211·89-s + 1.25·91-s − 0.609·97-s − 0.398·101-s − 0.985·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1152} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.958510323\)
\(L(\frac12)\) \(\approx\) \(1.958510323\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.732093454641320086632953897344, −8.788616180700843175291693061294, −8.396078230698006089481678801982, −7.31199108027851415208504538208, −6.32971218305453945925176285036, −5.76550563678950678115455651188, −4.27440644248673667608699010194, −3.95147465503398880690094499828, −2.30530051971351607466902904067, −1.16901968292393942798169217496, 1.16901968292393942798169217496, 2.30530051971351607466902904067, 3.95147465503398880690094499828, 4.27440644248673667608699010194, 5.76550563678950678115455651188, 6.32971218305453945925176285036, 7.31199108027851415208504538208, 8.396078230698006089481678801982, 8.788616180700843175291693061294, 9.732093454641320086632953897344

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