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Properties

Label 2-288-96.83-c1-0-13
Degree $2$
Conductor $288$
Sign $0.384 + 0.923i$
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  + (0.337 + 1.37i)2-s + (−1.77 + 0.928i)4-s + (−1.32 − 3.19i)5-s + (−2.32 − 2.32i)7-s + (−1.87 − 2.11i)8-s + (3.93 − 2.89i)10-s + (−1.47 − 3.55i)11-s + (4.49 + 1.86i)13-s + (2.41 − 3.98i)14-s + (2.27 − 3.28i)16-s − 4.93·17-s + (−1.98 + 4.79i)19-s + (5.30 + 4.42i)20-s + (4.37 − 3.21i)22-s + (−1.08 − 1.08i)23-s + ⋯
L(s)  = 1  + (0.238 + 0.971i)2-s + (−0.885 + 0.464i)4-s + (−0.591 − 1.42i)5-s + (−0.880 − 0.880i)7-s + (−0.662 − 0.749i)8-s + (1.24 − 0.915i)10-s + (−0.443 − 1.07i)11-s + (1.24 + 0.516i)13-s + (0.644 − 1.06i)14-s + (0.569 − 0.822i)16-s − 1.19·17-s + (−0.455 + 1.09i)19-s + (1.18 + 0.990i)20-s + (0.933 − 0.686i)22-s + (−0.227 − 0.227i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.384 + 0.923i$
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: \((2,\ 288,\ (\ :1/2),\ 0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624283 - 0.416311i\)
\(L(\frac12)\) \(\approx\) \(0.624283 - 0.416311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.337 - 1.37i)T \)
3 \( 1 \)
good5 \( 1 + (1.32 + 3.19i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.32 + 2.32i)T + 7iT^{2} \)
11 \( 1 + (1.47 + 3.55i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-4.49 - 1.86i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
19 \( 1 + (1.98 - 4.79i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.08 + 1.08i)T + 23iT^{2} \)
29 \( 1 + (-3.43 - 1.42i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 8.82iT - 31T^{2} \)
37 \( 1 + (-1.94 + 0.804i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.87 + 5.87i)T - 41iT^{2} \)
43 \( 1 + (2.44 - 1.01i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 1.61iT - 47T^{2} \)
53 \( 1 + (-5.62 + 2.32i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (7.67 - 3.17i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-3.16 + 7.65i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-3.31 - 1.37i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (2.13 - 2.13i)T - 71iT^{2} \)
73 \( 1 + (1.81 + 1.81i)T + 73iT^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 + (1.04 + 0.431i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-0.708 - 0.708i)T + 89iT^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−11.87601662936594198461172292658, −10.67041260923716067890661834558, −9.330560596689832370711718713783, −8.558498016664473227692928066651, −7.923463028780076932398557283070, −6.57675514686944741720198643868, −5.74922285286867964782839710563, −4.31929895907815831776106175530, −3.75017012858800716707488841683, −0.51868703733119623542674919936, 2.46843272947750395743286637086, 3.22095140096674258013779294297, 4.51881531103238109129013976035, 6.06119302632428565756191755257, 6.91769615649059870334879938733, 8.395798659729350437019062220789, 9.368944838075084701808465672304, 10.44878069308568438984242907333, 10.97677507768550494827931021002, 11.88431378593404502174830577241

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