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Properties

Label 2-288-32.13-c1-0-9
Degree $2$
Conductor $288$
Sign $0.983 - 0.182i$
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  + (−0.333 + 1.37i)2-s + (−1.77 − 0.916i)4-s + (1.20 − 0.498i)5-s + (2.59 − 2.59i)7-s + (1.85 − 2.13i)8-s + (0.283 + 1.82i)10-s + (−2.14 − 5.18i)11-s + (−0.984 − 0.407i)13-s + (2.69 + 4.43i)14-s + (2.31 + 3.25i)16-s + 0.979i·17-s + (5.68 + 2.35i)19-s + (−2.59 − 0.217i)20-s + (7.83 − 1.22i)22-s + (3.70 + 3.70i)23-s + ⋯
L(s)  = 1  + (−0.235 + 0.971i)2-s + (−0.888 − 0.458i)4-s + (0.538 − 0.223i)5-s + (0.980 − 0.980i)7-s + (0.655 − 0.755i)8-s + (0.0897 + 0.575i)10-s + (−0.647 − 1.56i)11-s + (−0.272 − 0.113i)13-s + (0.721 + 1.18i)14-s + (0.579 + 0.814i)16-s + 0.237i·17-s + (1.30 + 0.540i)19-s + (−0.580 − 0.0486i)20-s + (1.67 − 0.260i)22-s + (0.771 + 0.771i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.983 - 0.182i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.983 - 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20193 + 0.110770i\)
\(L(\frac12)\) \(\approx\) \(1.20193 + 0.110770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.333 - 1.37i)T \)
3 \( 1 \)
good5 \( 1 + (-1.20 + 0.498i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.59 + 2.59i)T - 7iT^{2} \)
11 \( 1 + (2.14 + 5.18i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.984 + 0.407i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 0.979iT - 17T^{2} \)
19 \( 1 + (-5.68 - 2.35i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.70 - 3.70i)T + 23iT^{2} \)
29 \( 1 + (-1.17 + 2.83i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.54T + 31T^{2} \)
37 \( 1 + (8.23 - 3.41i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.10 - 1.10i)T + 41iT^{2} \)
43 \( 1 + (3.47 + 8.37i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.15iT - 47T^{2} \)
53 \( 1 + (-2.55 - 6.16i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-8.95 + 3.70i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.00 - 4.84i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.14 + 2.76i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (10.0 - 10.0i)T - 71iT^{2} \)
73 \( 1 + (-8.11 - 8.11i)T + 73iT^{2} \)
79 \( 1 - 0.155iT - 79T^{2} \)
83 \( 1 + (5.13 + 2.12i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.15 + 6.15i)T - 89iT^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−11.70077590243138968932257409251, −10.66688513261908569358250862165, −9.906916209071632276880289487673, −8.725585700337955755975300701144, −7.936801482302265372396662862224, −7.12804255743706350898306407016, −5.69253153450401837490217681360, −5.13212309420609431652997933796, −3.63409321404641892189831502456, −1.14661228702317421506004904792, 1.86350126944493104921682849547, 2.77465513275185958205322515534, 4.70844028123895742707599765921, 5.28482563027680415054027546064, 7.12022615133385093240644930650, 8.158386913842760913847213975895, 9.204469477897133440438200958252, 9.924189734060448893351692006592, 10.84317197122316674498272735480, 11.85600590265987710263551867361

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