Properties

Degree $2$
Conductor $288$
Sign $-0.643 - 0.765i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.44 + 1.38i)2-s + (0.175 + 3.99i)4-s + (3.18 + 7.68i)5-s + (3.67 + 3.67i)7-s + (−5.27 + 6.01i)8-s + (−6.02 + 15.5i)10-s + (−6.10 − 14.7i)11-s + (2.82 − 6.80i)13-s + (0.228 + 10.3i)14-s + (−15.9 + 1.40i)16-s + 3.67i·17-s + (−1.65 − 0.686i)19-s + (−30.1 + 14.0i)20-s + (11.5 − 29.7i)22-s + (8.31 − 8.31i)23-s + ⋯
L(s)  = 1  + (0.722 + 0.691i)2-s + (0.0438 + 0.999i)4-s + (0.636 + 1.53i)5-s + (0.524 + 0.524i)7-s + (−0.659 + 0.752i)8-s + (−0.602 + 1.55i)10-s + (−0.554 − 1.33i)11-s + (0.216 − 0.523i)13-s + (0.0162 + 0.742i)14-s + (−0.996 + 0.0876i)16-s + 0.215i·17-s + (−0.0872 − 0.0361i)19-s + (−1.50 + 0.703i)20-s + (0.525 − 1.35i)22-s + (0.361 − 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.643 - 0.765i$
Motivic weight: \(2\)
Character: $\chi_{288} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.643 - 0.765i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08255 + 2.32349i\)
\(L(\frac12)\) \(\approx\) \(1.08255 + 2.32349i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.44 - 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (-3.18 - 7.68i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-3.67 - 3.67i)T + 49iT^{2} \)
11 \( 1 + (6.10 + 14.7i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-2.82 + 6.80i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 3.67iT - 289T^{2} \)
19 \( 1 + (1.65 + 0.686i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-8.31 + 8.31i)T - 529iT^{2} \)
29 \( 1 + (-38.8 - 16.0i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 4.11iT - 961T^{2} \)
37 \( 1 + (-19.8 - 47.9i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (21.1 + 21.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (0.102 + 0.247i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 39.3T + 2.20e3T^{2} \)
53 \( 1 + (22.6 - 9.36i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-101. + 41.9i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (14.0 + 5.81i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-3.67 + 8.87i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (75.7 + 75.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (29.0 + 29.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 2.76T + 6.24e3T^{2} \)
83 \( 1 + (-79.1 - 32.8i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (72.4 - 72.4i)T - 7.92e3iT^{2} \)
97 \( 1 - 66.0T + 9.40e3T^{2} \)
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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

−11.91973577534480932527012312792, −11.02134545692025192029400691531, −10.33485682755612768612057005818, −8.737883640300001848089359144458, −7.958854969754122862477091727583, −6.73938529424051219534258955090, −6.03369559451573580728541442687, −5.12569248230799767314891666967, −3.35615014554115323266195372206, −2.58113613187008232273344609275, 1.09548003901025944850315853619, 2.22628577750773223066949878013, 4.29782376605843941660410498698, 4.80373618685387254122731803199, 5.82185121494576177049789801143, 7.25645616878292101812267039488, 8.635130446731883692829856576041, 9.595349621386373371280495314043, 10.26603888424986103181091035174, 11.47191305329245337586756210187

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