Properties

Label 2-288-32.29-c1-0-14
Degree $2$
Conductor $288$
Sign $0.999 + 0.0142i$
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  + (1.36 + 0.387i)2-s + (1.69 + 1.05i)4-s + (−0.705 − 1.70i)5-s + (3.24 − 3.24i)7-s + (1.90 + 2.09i)8-s + (−0.299 − 2.59i)10-s + (−3.38 + 1.40i)11-s + (−0.503 + 1.21i)13-s + (5.66 − 3.15i)14-s + (1.77 + 3.58i)16-s − 0.622i·17-s + (−2.14 + 5.17i)19-s + (0.597 − 3.63i)20-s + (−5.15 + 0.595i)22-s + (2.47 + 2.47i)23-s + ⋯
L(s)  = 1  + (0.961 + 0.274i)2-s + (0.849 + 0.527i)4-s + (−0.315 − 0.762i)5-s + (1.22 − 1.22i)7-s + (0.672 + 0.740i)8-s + (−0.0946 − 0.819i)10-s + (−1.02 + 0.423i)11-s + (−0.139 + 0.337i)13-s + (1.51 − 0.842i)14-s + (0.443 + 0.896i)16-s − 0.151i·17-s + (−0.491 + 1.18i)19-s + (0.133 − 0.813i)20-s + (−1.09 + 0.126i)22-s + (0.516 + 0.516i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.999 + 0.0142i$
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: \((2,\ 288,\ (\ :1/2),\ 0.999 + 0.0142i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26367 - 0.0161459i\)
\(L(\frac12)\) \(\approx\) \(2.26367 - 0.0161459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.387i)T \)
3 \( 1 \)
good5 \( 1 + (0.705 + 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-3.24 + 3.24i)T - 7iT^{2} \)
11 \( 1 + (3.38 - 1.40i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.503 - 1.21i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.622iT - 17T^{2} \)
19 \( 1 + (2.14 - 5.17i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.47 - 2.47i)T + 23iT^{2} \)
29 \( 1 + (-2.16 - 0.897i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + (0.0714 + 0.172i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (8.50 + 8.50i)T + 41iT^{2} \)
43 \( 1 + (3.62 - 1.50i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 5.02iT - 47T^{2} \)
53 \( 1 + (-7.15 + 2.96i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.52 - 3.68i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (3.07 + 1.27i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (2.17 + 0.901i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-1.11 + 1.11i)T - 71iT^{2} \)
73 \( 1 + (-3.71 - 3.71i)T + 73iT^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 + (-4.69 + 11.3i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-3.54 + 3.54i)T - 89iT^{2} \)
97 \( 1 - 0.139T + 97T^{2} \)
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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

−12.00242104679676393555518087619, −10.99792222055030549557737611905, −10.31176628237177427377469608787, −8.558513835304037929048355065140, −7.73890347957140676539856784395, −7.06122581864060978955384324324, −5.39173026194271342338049459822, −4.69593144706747593136435969308, −3.76146993530688029531580273545, −1.78238814239739675849689798507, 2.20646189003346243800940611575, 3.15620733850910241761375118850, 4.83548056133472947278952599954, 5.47909525932750287781332103271, 6.74782309690888576803643630306, 7.83030380822134560050156369838, 8.864015344191074854340032330619, 10.46402894156948380323659989669, 11.08154720588103417987056905028, 11.71231269346941758235395054478

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