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Properties

Label 2-1152-8.3-c2-0-9
Degree $2$
Conductor $1152$
Sign $-0.707 - 0.707i$
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  + 5.29i·5-s + 7.48i·7-s − 5.65·11-s − 4i·13-s + 21.1·17-s + 29.9·19-s + 22.6i·23-s − 3.00·25-s + 5.29i·29-s + 22.4i·31-s − 39.5·35-s + 28i·37-s − 63.4·41-s − 29.9·43-s + 67.8i·47-s + ⋯
L(s)  = 1  + 1.05i·5-s + 1.06i·7-s − 0.514·11-s − 0.307i·13-s + 1.24·17-s + 1.57·19-s + 0.983i·23-s − 0.120·25-s + 0.182i·29-s + 0.724i·31-s − 1.13·35-s + 0.756i·37-s − 1.54·41-s − 0.696·43-s + 1.44i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.707 - 0.707i$
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: \((2,\ 1152,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.594088580\)
\(L(\frac12)\) \(\approx\) \(1.594088580\)
\(L(2)\) 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 \( 1 - 5.29iT - 25T^{2} \)
7 \( 1 - 7.48iT - 49T^{2} \)
11 \( 1 + 5.65T + 121T^{2} \)
13 \( 1 + 4iT - 169T^{2} \)
17 \( 1 - 21.1T + 289T^{2} \)
19 \( 1 - 29.9T + 361T^{2} \)
23 \( 1 - 22.6iT - 529T^{2} \)
29 \( 1 - 5.29iT - 841T^{2} \)
31 \( 1 - 22.4iT - 961T^{2} \)
37 \( 1 - 28iT - 1.36e3T^{2} \)
41 \( 1 + 63.4T + 1.68e3T^{2} \)
43 \( 1 + 29.9T + 1.84e3T^{2} \)
47 \( 1 - 67.8iT - 2.20e3T^{2} \)
53 \( 1 + 47.6iT - 2.80e3T^{2} \)
59 \( 1 + 101.T + 3.48e3T^{2} \)
61 \( 1 + 76iT - 3.72e3T^{2} \)
67 \( 1 + 59.8T + 4.48e3T^{2} \)
71 \( 1 - 90.5iT - 5.04e3T^{2} \)
73 \( 1 - 26T + 5.32e3T^{2} \)
79 \( 1 + 127. iT - 6.24e3T^{2} \)
83 \( 1 + 118.T + 6.88e3T^{2} \)
89 \( 1 - 42.3T + 7.92e3T^{2} \)
97 \( 1 - 18T + 9.40e3T^{2} \)
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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

−9.944955484336995017149725676837, −9.225208863624637151978879813585, −8.121708168482759198622319381594, −7.50363955038041796993526542031, −6.58304988121189354294533976721, −5.60121560605486651906338041518, −5.06589186866292263392870223090, −3.19064071812404276465968768338, −3.06523072752862945526910690953, −1.52540446504171779292672882311, 0.50650366440550799180410077335, 1.45558348571195851950652241976, 3.07998819732355007396660532401, 4.11778201145709046732612397744, 4.98702103895291392051582459642, 5.70957114728700256307327981516, 6.99015529070651738368660371161, 7.67919450936635917566454324596, 8.420556881709538013301723881579, 9.358907380544258337530650629082

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