Properties

Label 2-288-96.83-c1-0-5
Degree $2$
Conductor $288$
Sign $0.577 - 0.816i$
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  + (−0.290 + 1.38i)2-s + (−1.83 − 0.804i)4-s + (−1.12 − 2.71i)5-s + (3.03 + 3.03i)7-s + (1.64 − 2.29i)8-s + (4.08 − 0.766i)10-s + (0.616 + 1.48i)11-s + (3.35 + 1.38i)13-s + (−5.07 + 3.31i)14-s + (2.70 + 2.94i)16-s + 4.76·17-s + (1.13 − 2.73i)19-s + (−0.125 + 5.87i)20-s + (−2.24 + 0.420i)22-s + (4.11 + 4.11i)23-s + ⋯
L(s)  = 1  + (−0.205 + 0.978i)2-s + (−0.915 − 0.402i)4-s + (−0.502 − 1.21i)5-s + (1.14 + 1.14i)7-s + (0.582 − 0.813i)8-s + (1.29 − 0.242i)10-s + (0.185 + 0.449i)11-s + (0.929 + 0.385i)13-s + (−1.35 + 0.885i)14-s + (0.676 + 0.736i)16-s + 1.15·17-s + (0.259 − 0.627i)19-s + (−0.0281 + 1.31i)20-s + (−0.477 + 0.0896i)22-s + (0.857 + 0.857i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02677 + 0.530995i\)
\(L(\frac12)\) \(\approx\) \(1.02677 + 0.530995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.290 - 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (1.12 + 2.71i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-3.03 - 3.03i)T + 7iT^{2} \)
11 \( 1 + (-0.616 - 1.48i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-3.35 - 1.38i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 + (-1.13 + 2.73i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.11 - 4.11i)T + 23iT^{2} \)
29 \( 1 + (8.16 + 3.38i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 6.16iT - 31T^{2} \)
37 \( 1 + (9.38 - 3.88i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.169 - 0.169i)T - 41iT^{2} \)
43 \( 1 + (-7.57 + 3.13i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.44iT - 47T^{2} \)
53 \( 1 + (-1.99 + 0.824i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.21 + 0.503i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.04 + 2.52i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-3.91 - 1.62i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.28 - 5.28i)T - 71iT^{2} \)
73 \( 1 + (1.57 + 1.57i)T + 73iT^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + (-3.31 - 1.37i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.99 - 6.99i)T + 89iT^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−11.99562018099738649703822829093, −11.20383978607189357329669657568, −9.536509268683104282828635505189, −8.882208680826250646809524447656, −8.190181978099401560866511904598, −7.33413962497752768586909656261, −5.70206592459844012300922610419, −5.13994102031073485083246346467, −4.04180310355489477111123126901, −1.40817960757311684046190122787, 1.30266992917119704854418418550, 3.23686721499692778979716723598, 3.91910891123112701320891392184, 5.38602838304049763993970039815, 7.13242654227856837539662935863, 7.84808345242215376130008315242, 8.816118878773385147164498006467, 10.36144264845868112228818827321, 10.75896558820798717134492967824, 11.31100260165952433468545529149

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