Properties

Label 2-1152-1.1-c3-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s − 10·7-s + 68·11-s − 46·13-s + 74·17-s − 16·19-s + 20·23-s − 61·25-s − 228·29-s − 162·31-s + 80·35-s + 262·37-s − 30·41-s − 264·43-s − 124·47-s − 243·49-s + 204·53-s − 544·55-s + 340·59-s + 950·61-s + 368·65-s + 436·67-s + 780·71-s + 518·73-s − 680·77-s − 1.01e3·79-s + 852·83-s + ⋯
L(s)  = 1  − 0.715·5-s − 0.539·7-s + 1.86·11-s − 0.981·13-s + 1.05·17-s − 0.193·19-s + 0.181·23-s − 0.487·25-s − 1.45·29-s − 0.938·31-s + 0.386·35-s + 1.16·37-s − 0.114·41-s − 0.936·43-s − 0.384·47-s − 0.708·49-s + 0.528·53-s − 1.33·55-s + 0.750·59-s + 1.99·61-s + 0.702·65-s + 0.795·67-s + 1.30·71-s + 0.830·73-s − 1.00·77-s − 1.43·79-s + 1.12·83-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: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.557281876\)
\(L(\frac12)\) \(\approx\) \(1.557281876\)
\(L(\frac{5}{2})\) 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 \( 1 + 8 T + p^{3} T^{2} \)
7 \( 1 + 10 T + p^{3} T^{2} \)
11 \( 1 - 68 T + p^{3} T^{2} \)
13 \( 1 + 46 T + p^{3} T^{2} \)
17 \( 1 - 74 T + p^{3} T^{2} \)
19 \( 1 + 16 T + p^{3} T^{2} \)
23 \( 1 - 20 T + p^{3} T^{2} \)
29 \( 1 + 228 T + p^{3} T^{2} \)
31 \( 1 + 162 T + p^{3} T^{2} \)
37 \( 1 - 262 T + p^{3} T^{2} \)
41 \( 1 + 30 T + p^{3} T^{2} \)
43 \( 1 + 264 T + p^{3} T^{2} \)
47 \( 1 + 124 T + p^{3} T^{2} \)
53 \( 1 - 204 T + p^{3} T^{2} \)
59 \( 1 - 340 T + p^{3} T^{2} \)
61 \( 1 - 950 T + p^{3} T^{2} \)
67 \( 1 - 436 T + p^{3} T^{2} \)
71 \( 1 - 780 T + p^{3} T^{2} \)
73 \( 1 - 518 T + p^{3} T^{2} \)
79 \( 1 + 1010 T + p^{3} T^{2} \)
83 \( 1 - 852 T + p^{3} T^{2} \)
89 \( 1 - 686 T + p^{3} T^{2} \)
97 \( 1 + 806 T + p^{3} T^{2} \)
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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

−9.584768619799777084498610078480, −8.647629104418965618815140603917, −7.66168347747263511819648736303, −7.00748509480634970138410699907, −6.16060259970194816228913092067, −5.12298431025186768376372305364, −3.91449221006273475787511071876, −3.48807123702819313766786860084, −1.97808567420858975024116395745, −0.64405930285397203288250744345, 0.64405930285397203288250744345, 1.97808567420858975024116395745, 3.48807123702819313766786860084, 3.91449221006273475787511071876, 5.12298431025186768376372305364, 6.16060259970194816228913092067, 7.00748509480634970138410699907, 7.66168347747263511819648736303, 8.647629104418965618815140603917, 9.584768619799777084498610078480

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