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Properties

Label 2-288-32.13-c1-0-15
Degree $2$
Conductor $288$
Sign $0.998 + 0.0541i$
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  + (1.41 − 0.0402i)2-s + (1.99 − 0.113i)4-s + (1.51 − 0.626i)5-s + (−1.32 + 1.32i)7-s + (2.81 − 0.241i)8-s + (2.11 − 0.946i)10-s + (0.938 + 2.26i)11-s + (−4.73 − 1.96i)13-s + (−1.82 + 1.92i)14-s + (3.97 − 0.454i)16-s − 5.78i·17-s + (−1.04 − 0.431i)19-s + (2.94 − 1.42i)20-s + (1.41 + 3.16i)22-s + (4.29 + 4.29i)23-s + ⋯
L(s)  = 1  + (0.999 − 0.0284i)2-s + (0.998 − 0.0568i)4-s + (0.676 − 0.280i)5-s + (−0.500 + 0.500i)7-s + (0.996 − 0.0852i)8-s + (0.668 − 0.299i)10-s + (0.283 + 0.683i)11-s + (−1.31 − 0.543i)13-s + (−0.486 + 0.514i)14-s + (0.993 − 0.113i)16-s − 1.40i·17-s + (−0.239 − 0.0990i)19-s + (0.659 − 0.318i)20-s + (0.302 + 0.675i)22-s + (0.895 + 0.895i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.998 + 0.0541i$
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: \((2,\ 288,\ (\ :1/2),\ 0.998 + 0.0541i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39236 - 0.0647935i\)
\(L(\frac12)\) \(\approx\) \(2.39236 - 0.0647935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0402i)T \)
3 \( 1 \)
good5 \( 1 + (-1.51 + 0.626i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.32 - 1.32i)T - 7iT^{2} \)
11 \( 1 + (-0.938 - 2.26i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (4.73 + 1.96i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 5.78iT - 17T^{2} \)
19 \( 1 + (1.04 + 0.431i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.29 - 4.29i)T + 23iT^{2} \)
29 \( 1 + (0.389 - 0.940i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 7.43T + 31T^{2} \)
37 \( 1 + (3.67 - 1.52i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.474 - 0.474i)T + 41iT^{2} \)
43 \( 1 + (0.409 + 0.987i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 2.73iT - 47T^{2} \)
53 \( 1 + (4.55 + 10.9i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (8.68 - 3.59i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.48 + 3.58i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-6.11 + 14.7i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (10.5 - 10.5i)T - 71iT^{2} \)
73 \( 1 + (-6.86 - 6.86i)T + 73iT^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + (-15.2 - 6.30i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.07 + 6.07i)T - 89iT^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−12.09847461876045157958051193555, −11.10313327288380196166873191247, −9.817991542222813413777285583732, −9.312605806929509625806774364570, −7.54151358548018374477223803001, −6.79062185436123555886743217178, −5.46533818366016352590732031046, −4.92923685192959067072655983915, −3.25357191494106228544427479867, −2.07397518715854664325664340959, 2.07041353149387267723044139409, 3.43094923767502322143328039397, 4.59576979009122131274169781359, 5.91138378238779425114833989673, 6.61046481187186936935462240843, 7.61306412184283134722752654707, 9.075231891889680999113658324467, 10.28434319303190266827222297468, 10.83182489931532126198266404382, 12.06816643118825624056207703040

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