Properties

Degree $2$
Conductor $1152$
Sign $0.0805 - 0.996i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.00 + 1.00i)5-s − 10.0·7-s + (−2.26 + 2.26i)11-s + (6.88 − 6.88i)13-s + 22.3·17-s + (−16.8 − 16.8i)19-s + 33.2·23-s − 22.9i·25-s + (−24.6 + 24.6i)29-s + 41.3i·31-s + (−10.1 − 10.1i)35-s + (6.60 + 6.60i)37-s + 47.1i·41-s + (−48.8 + 48.8i)43-s + 45.6i·47-s + ⋯
L(s)  = 1  + (0.201 + 0.201i)5-s − 1.43·7-s + (−0.205 + 0.205i)11-s + (0.529 − 0.529i)13-s + 1.31·17-s + (−0.889 − 0.889i)19-s + 1.44·23-s − 0.918i·25-s + (−0.849 + 0.849i)29-s + 1.33i·31-s + (−0.288 − 0.288i)35-s + (0.178 + 0.178i)37-s + 1.14i·41-s + (−1.13 + 1.13i)43-s + 0.970i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.0805 - 0.996i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.0805 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.199093163\)
\(L(\frac12)\) \(\approx\) \(1.199093163\)
\(L(2)\) 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 \( 1 + (-1.00 - 1.00i)T + 25iT^{2} \)
7 \( 1 + 10.0T + 49T^{2} \)
11 \( 1 + (2.26 - 2.26i)T - 121iT^{2} \)
13 \( 1 + (-6.88 + 6.88i)T - 169iT^{2} \)
17 \( 1 - 22.3T + 289T^{2} \)
19 \( 1 + (16.8 + 16.8i)T + 361iT^{2} \)
23 \( 1 - 33.2T + 529T^{2} \)
29 \( 1 + (24.6 - 24.6i)T - 841iT^{2} \)
31 \( 1 - 41.3iT - 961T^{2} \)
37 \( 1 + (-6.60 - 6.60i)T + 1.36e3iT^{2} \)
41 \( 1 - 47.1iT - 1.68e3T^{2} \)
43 \( 1 + (48.8 - 48.8i)T - 1.84e3iT^{2} \)
47 \( 1 - 45.6iT - 2.20e3T^{2} \)
53 \( 1 + (-25.1 - 25.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (6.23 - 6.23i)T - 3.48e3iT^{2} \)
61 \( 1 + (35.9 - 35.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (-10.2 - 10.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 11.9T + 5.04e3T^{2} \)
73 \( 1 - 111. iT - 5.32e3T^{2} \)
79 \( 1 + 4.46iT - 6.24e3T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 21.9iT - 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−9.836178228200111464885711563195, −9.082182963719328497282932414815, −8.234523482187181629983167018199, −7.12271402609655725662825372472, −6.51854106021547323045873062426, −5.68739997258243519520975719594, −4.66181569304245060865440438635, −3.29506733452513610155094726730, −2.85329541836955151578591311828, −1.09848775130838655116124859275, 0.40766975970569076196772846249, 1.92773032884527867235246043571, 3.30060306207186941847531244265, 3.87933238064782241536357055773, 5.35556324797926896246801486790, 6.00663700084335268464161517042, 6.84023661609311831268433434563, 7.72519847322499924820932848020, 8.763276485667331114851837337257, 9.448222796531106253747622887196

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