Properties

Label 2-288-32.5-c1-0-5
Degree $2$
Conductor $288$
Sign $0.792 - 0.610i$
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  + (−1.40 + 0.129i)2-s + (1.96 − 0.366i)4-s + (2.51 + 1.04i)5-s + (2.01 + 2.01i)7-s + (−2.72 + 0.771i)8-s + (−3.67 − 1.13i)10-s + (1.32 − 3.20i)11-s + (−5.55 + 2.30i)13-s + (−3.09 − 2.56i)14-s + (3.73 − 1.43i)16-s + 4.16i·17-s + (4.49 − 1.86i)19-s + (5.32 + 1.12i)20-s + (−1.45 + 4.68i)22-s + (1.94 − 1.94i)23-s + ⋯
L(s)  = 1  + (−0.995 + 0.0919i)2-s + (0.983 − 0.183i)4-s + (1.12 + 0.465i)5-s + (0.759 + 0.759i)7-s + (−0.962 + 0.272i)8-s + (−1.16 − 0.360i)10-s + (0.400 − 0.966i)11-s + (−1.54 + 0.638i)13-s + (−0.826 − 0.686i)14-s + (0.932 − 0.359i)16-s + 1.01i·17-s + (1.03 − 0.427i)19-s + (1.19 + 0.251i)20-s + (−0.309 + 0.998i)22-s + (0.405 − 0.405i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.792 - 0.610i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.792 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00346 + 0.341800i\)
\(L(\frac12)\) \(\approx\) \(1.00346 + 0.341800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.129i)T \)
3 \( 1 \)
good5 \( 1 + (-2.51 - 1.04i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.01 - 2.01i)T + 7iT^{2} \)
11 \( 1 + (-1.32 + 3.20i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (5.55 - 2.30i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
19 \( 1 + (-4.49 + 1.86i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.94 + 1.94i)T - 23iT^{2} \)
29 \( 1 + (-1.96 - 4.73i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 2.87T + 31T^{2} \)
37 \( 1 + (-6.31 - 2.61i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.756 - 0.756i)T - 41iT^{2} \)
43 \( 1 + (-1.53 + 3.71i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 1.08iT - 47T^{2} \)
53 \( 1 + (-2.90 + 7.01i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (10.3 + 4.28i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (2.97 + 7.18i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.88 + 9.38i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-1.88 - 1.88i)T + 71iT^{2} \)
73 \( 1 + (7.00 - 7.00i)T - 73iT^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 + (-2.30 + 0.954i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (7.65 + 7.65i)T + 89iT^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−11.62441659258822142405579129224, −10.87072643523273066610650062098, −9.844318987857345735095257764771, −9.163252128493728511728427710260, −8.262910576473350203573292579769, −7.06158786421230155169145803450, −6.11669570354765528288412800702, −5.16843044175304245974278598780, −2.84138277330583506857636408443, −1.75171129685806177230919323592, 1.27921614885884351261659353237, 2.58770769303474819131985411235, 4.67513731113653168748819883622, 5.76222174652286010078835535082, 7.31931764910861922973148658321, 7.63392631444194750830869352338, 9.246565624562903402098875767768, 9.697609647321595802032058282756, 10.44194897308167367120962740732, 11.64665966097721223118901017582

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