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Properties

Label 2-288-32.29-c1-0-9
Degree $2$
Conductor $288$
Sign $0.710 - 0.704i$
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  + (1.40 + 0.167i)2-s + (1.94 + 0.470i)4-s + (0.707 + 1.70i)5-s + (−2.74 + 2.74i)7-s + (2.65 + 0.985i)8-s + (0.707 + 2.51i)10-s + (−0.135 + 0.0560i)11-s + (1.18 − 2.85i)13-s + (−4.32 + 3.40i)14-s + (3.55 + 1.82i)16-s − 6.44i·17-s + (0.805 − 1.94i)19-s + (0.571 + 3.65i)20-s + (−0.199 + 0.0560i)22-s + (−0.749 − 0.749i)23-s + ⋯
L(s)  = 1  + (0.992 + 0.118i)2-s + (0.971 + 0.235i)4-s + (0.316 + 0.763i)5-s + (−1.03 + 1.03i)7-s + (0.937 + 0.348i)8-s + (0.223 + 0.795i)10-s + (−0.0408 + 0.0169i)11-s + (0.327 − 0.790i)13-s + (−1.15 + 0.908i)14-s + (0.889 + 0.457i)16-s − 1.56i·17-s + (0.184 − 0.445i)19-s + (0.127 + 0.816i)20-s + (−0.0425 + 0.0119i)22-s + (−0.156 − 0.156i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.710 - 0.704i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.710 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07540 + 0.854344i\)
\(L(\frac12)\) \(\approx\) \(2.07540 + 0.854344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.167i)T \)
3 \( 1 \)
good5 \( 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.74 - 2.74i)T - 7iT^{2} \)
11 \( 1 + (0.135 - 0.0560i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.18 + 2.85i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.44iT - 17T^{2} \)
19 \( 1 + (-0.805 + 1.94i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.749 + 0.749i)T + 23iT^{2} \)
29 \( 1 + (4.32 + 1.79i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (-1.73 - 4.18i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.49 - 2.49i)T + 41iT^{2} \)
43 \( 1 + (6.10 - 2.52i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.66iT - 47T^{2} \)
53 \( 1 + (1.64 - 0.682i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.43 + 3.47i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (3.46 + 1.43i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (14.0 + 5.83i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-3.40 + 3.40i)T - 71iT^{2} \)
73 \( 1 + (0.442 + 0.442i)T + 73iT^{2} \)
79 \( 1 - 7.07iT - 79T^{2} \)
83 \( 1 + (2.99 - 7.23i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.21 + 4.21i)T - 89iT^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−12.03526596845764263982398333377, −11.23056759697113292444149193189, −10.17137654863614522272949270604, −9.227866553197712331543422700423, −7.78193154552871334085420666282, −6.66308628609074644450148298920, −6.00141764304122725010045128828, −4.94527341390506713238655212494, −3.20374834788595716431735964001, −2.60884822454516106776966240064, 1.58740242097832837673522410638, 3.51217662958108961341824946449, 4.28721634869839379405815941781, 5.66528020989761744944015480842, 6.52231718111350978835705899887, 7.53645542044266062080830067920, 8.948319124319717679709438669539, 10.06903883068797716889164819913, 10.77176833474274350398555626721, 11.94659080905821666244046297708

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