Properties

Label 2-1152-8.3-c2-0-4
Degree $2$
Conductor $1152$
Sign $-i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·5-s + 24i·13-s − 30·17-s − 39·25-s + 40i·29-s − 24i·37-s − 18·41-s + 49·49-s + 56i·53-s + 120i·61-s + 192·65-s + 110·73-s + 240i·85-s − 78·89-s − 130·97-s + ⋯
L(s)  = 1  − 1.60i·5-s + 1.84i·13-s − 1.76·17-s − 1.56·25-s + 1.37i·29-s − 0.648i·37-s − 0.439·41-s + 0.999·49-s + 1.05i·53-s + 1.96i·61-s + 2.95·65-s + 1.50·73-s + 2.82i·85-s − 0.876·89-s − 1.34·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7952995811\)
\(L(\frac12)\) \(\approx\) \(0.7952995811\)
\(L(2)\) 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 \( 1 + 8iT - 25T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 24iT - 169T^{2} \)
17 \( 1 + 30T + 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 40iT - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 24iT - 1.36e3T^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 56iT - 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 120iT - 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 110T + 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + 78T + 7.92e3T^{2} \)
97 \( 1 + 130T + 9.40e3T^{2} \)
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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

−9.429589192599625196528852247727, −8.947412239860135317690157458352, −8.553712908357946037454445823122, −7.27239906329365288191992880793, −6.52076613214452897279303420803, −5.40304349924919107735329091352, −4.53456956091277941066100927246, −4.04099505427912534838183770285, −2.24595443894314882882760091766, −1.27631972189360007629408374547, 0.23927286703211987287071772735, 2.24300894325048761371351860277, 2.98829560410265368510239700151, 3.94296339704737778707105454369, 5.21284316672106096258114447173, 6.24682051394766786845046640549, 6.79589879245493768838905554065, 7.73816542994360663352345466260, 8.400380869890373878263185751725, 9.637129277635446993759658749884

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