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Properties

Label 2-288-32.21-c1-0-14
Degree $2$
Conductor $288$
Sign $-0.498 + 0.867i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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  + (−1.39 − 0.202i)2-s + (1.91 + 0.566i)4-s + (1.53 − 3.71i)5-s + (−1.56 − 1.56i)7-s + (−2.57 − 1.18i)8-s + (−2.90 + 4.88i)10-s + (−3.83 − 1.59i)11-s + (1.23 + 2.98i)13-s + (1.87 + 2.50i)14-s + (3.35 + 2.17i)16-s + 3.82i·17-s + (−2.98 − 7.20i)19-s + (5.05 − 6.25i)20-s + (5.05 + 3.00i)22-s + (−0.793 + 0.793i)23-s + ⋯
L(s)  = 1  + (−0.989 − 0.143i)2-s + (0.959 + 0.283i)4-s + (0.687 − 1.66i)5-s + (−0.590 − 0.590i)7-s + (−0.908 − 0.417i)8-s + (−0.918 + 1.54i)10-s + (−1.15 − 0.479i)11-s + (0.342 + 0.826i)13-s + (0.500 + 0.669i)14-s + (0.839 + 0.543i)16-s + 0.927i·17-s + (−0.685 − 1.65i)19-s + (1.12 − 1.39i)20-s + (1.07 + 0.640i)22-s + (−0.165 + 0.165i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.498 + 0.867i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.498 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360987 - 0.623638i\)
\(L(\frac12)\) \(\approx\) \(0.360987 - 0.623638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.202i)T \)
3 \( 1 \)
good5 \( 1 + (-1.53 + 3.71i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.56 + 1.56i)T + 7iT^{2} \)
11 \( 1 + (3.83 + 1.59i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.23 - 2.98i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 3.82iT - 17T^{2} \)
19 \( 1 + (2.98 + 7.20i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.793 - 0.793i)T - 23iT^{2} \)
29 \( 1 + (-1.97 + 0.819i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 2.27T + 31T^{2} \)
37 \( 1 + (-2.48 + 6.00i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.72 + 3.72i)T - 41iT^{2} \)
43 \( 1 + (-6.59 - 2.73i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 2.53iT - 47T^{2} \)
53 \( 1 + (-5.60 - 2.32i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.52 + 6.09i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.34 - 1.38i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-9.88 + 4.09i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-2.08 - 2.08i)T + 71iT^{2} \)
73 \( 1 + (3.11 - 3.11i)T - 73iT^{2} \)
79 \( 1 - 3.53iT - 79T^{2} \)
83 \( 1 + (-6.34 - 15.3i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (6.94 + 6.94i)T + 89iT^{2} \)
97 \( 1 + 6.02T + 97T^{2} \)
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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

−11.25502679190727986665757387018, −10.40670186524451242284717453870, −9.431467641811199539362058983855, −8.802232687118779906930022449351, −7.980140449275163149705378917223, −6.63933018907835058389241101050, −5.60959556926948721941131902336, −4.20071912297952064773209335576, −2.30866893311729689305532521429, −0.70903099371488302273445166662, 2.33788472572079377102476823128, 3.07232598897638963936671625143, 5.66028814837425571054866200237, 6.30736072672027894390939606825, 7.33204036767496159612489451126, 8.198655876806091612822584154229, 9.585102126377511518122691272816, 10.25997997030673153457609165129, 10.68987077207440557955157626211, 11.87958688268716504643505961088

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