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Properties

Label 2-1152-12.11-c1-0-7
Degree $2$
Conductor $1152$
Sign $0.816 - 0.577i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.585i·5-s + 0.828i·7-s + 2.82·11-s + 2.82·13-s + 2.58i·17-s − 5.65i·19-s − 6.82·23-s + 4.65·25-s + 3.41i·29-s + 8.82i·31-s − 0.485·35-s + 7.65·37-s + 5.41i·41-s − 1.65i·43-s + 4.48·47-s + ⋯
L(s)  = 1  + 0.261i·5-s + 0.313i·7-s + 0.852·11-s + 0.784·13-s + 0.627i·17-s − 1.29i·19-s − 1.42·23-s + 0.931·25-s + 0.634i·29-s + 1.58i·31-s − 0.0820·35-s + 1.25·37-s + 0.845i·41-s − 0.252i·43-s + 0.654·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.728452146\)
\(L(\frac12)\) \(\approx\) \(1.728452146\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585iT - 5T^{2} \)
7 \( 1 - 0.828iT - 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 2.58iT - 17T^{2} \)
19 \( 1 + 5.65iT - 19T^{2} \)
23 \( 1 + 6.82T + 23T^{2} \)
29 \( 1 - 3.41iT - 29T^{2} \)
31 \( 1 - 8.82iT - 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 - 5.41iT - 41T^{2} \)
43 \( 1 + 1.65iT - 43T^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 - 9.07iT - 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 - 3.65T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 - 3.75iT - 89T^{2} \)
97 \( 1 - 2.34T + 97T^{2} \)
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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

−9.879702338006211502008750907890, −8.882022699604063161175003990171, −8.501076689446542083857572270306, −7.27706027875578380231569869976, −6.52331597852173763444731355765, −5.79265631808348482552702571428, −4.62728601826417642329631424696, −3.70263642532890821053294044803, −2.62112419134848905602478919957, −1.24760914673966060729512368004, 0.914404346678317626253285303243, 2.24933474513764952603908938743, 3.76694970362761823829882908992, 4.25328063149764597039923097418, 5.64686861246427816297408382419, 6.24315695442239924398328202053, 7.29447520560921211994125246126, 8.119663144162866958006445141190, 8.875362810926045365395798507057, 9.776044519940063914114074583613

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