Properties

Degree $2$
Conductor $288$
Sign $-0.922 + 0.385i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.46 − 1.35i)2-s + (0.309 − 3.98i)4-s + (−1.74 − 4.21i)5-s + (−0.392 − 0.392i)7-s + (−4.96 − 6.27i)8-s + (−8.29 − 3.81i)10-s + (−2.90 − 7.02i)11-s + (−4.50 + 10.8i)13-s + (−1.10 − 0.0429i)14-s + (−15.8 − 2.46i)16-s − 10.5i·17-s + (−1.88 − 0.781i)19-s + (−17.3 + 5.66i)20-s + (−13.8 − 6.35i)22-s + (0.445 − 0.445i)23-s + ⋯
L(s)  = 1  + (0.733 − 0.679i)2-s + (0.0772 − 0.997i)4-s + (−0.349 − 0.843i)5-s + (−0.0560 − 0.0560i)7-s + (−0.620 − 0.784i)8-s + (−0.829 − 0.381i)10-s + (−0.264 − 0.638i)11-s + (−0.346 + 0.836i)13-s + (−0.0792 − 0.00306i)14-s + (−0.988 − 0.154i)16-s − 0.620i·17-s + (−0.0993 − 0.0411i)19-s + (−0.867 + 0.283i)20-s + (−0.627 − 0.288i)22-s + (0.0193 − 0.0193i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.922 + 0.385i$
Motivic weight: \(2\)
Character: $\chi_{288} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.922 + 0.385i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.352360 - 1.75762i\)
\(L(\frac12)\) \(\approx\) \(0.352360 - 1.75762i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.46 + 1.35i)T \)
3 \( 1 \)
good5 \( 1 + (1.74 + 4.21i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (0.392 + 0.392i)T + 49iT^{2} \)
11 \( 1 + (2.90 + 7.02i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (4.50 - 10.8i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 10.5iT - 289T^{2} \)
19 \( 1 + (1.88 + 0.781i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-0.445 + 0.445i)T - 529iT^{2} \)
29 \( 1 + (0.741 + 0.307i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 47.6iT - 961T^{2} \)
37 \( 1 + (-14.5 - 35.0i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-11.3 - 11.3i)T + 1.68e3iT^{2} \)
43 \( 1 + (14.6 + 35.3i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 80.5T + 2.20e3T^{2} \)
53 \( 1 + (-66.6 + 27.5i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (65.0 - 26.9i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-87.4 - 36.2i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (7.12 - 17.1i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-14.8 - 14.8i)T + 5.04e3iT^{2} \)
73 \( 1 + (-18.6 - 18.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 36.2T + 6.24e3T^{2} \)
83 \( 1 + (27.0 + 11.2i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-56.4 + 56.4i)T - 7.92e3iT^{2} \)
97 \( 1 - 158.T + 9.40e3T^{2} \)
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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

−11.54812300303160231090954668244, −10.38535214139185880900518277051, −9.399049423876149679653253687569, −8.516223474731339903815874185344, −7.11394134539220680204048064086, −5.85863026508366736916326679149, −4.81973529340321647612178365970, −3.91529463728286543966543137547, −2.42389742006015577791181858122, −0.69328973189879551980210672509, 2.60783065799370404944910938443, 3.70491237039488386176446026300, 4.96650959039967035387763242600, 6.07145841549530974317933369536, 7.14249804325180605683784630318, 7.76039984227103016054544910456, 8.923407576328691213973490316694, 10.32795422287034058979922060027, 11.11390161971967279089344749733, 12.33754101738687634118914099223

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