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Properties

Label 2-288-96.11-c1-0-13
Degree $2$
Conductor $288$
Sign $0.618 + 0.785i$
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  + (1.16 − 0.797i)2-s + (0.727 − 1.86i)4-s + (1.65 + 0.686i)5-s + (−0.456 + 0.456i)7-s + (−0.636 − 2.75i)8-s + (2.48 − 0.520i)10-s + (2.79 + 1.15i)11-s + (−1.29 − 3.12i)13-s + (−0.168 + 0.897i)14-s + (−2.94 − 2.70i)16-s + 3.55·17-s + (−5.61 + 2.32i)19-s + (2.48 − 2.58i)20-s + (4.18 − 0.876i)22-s + (−4.10 + 4.10i)23-s + ⋯
L(s)  = 1  + (0.825 − 0.564i)2-s + (0.363 − 0.931i)4-s + (0.740 + 0.306i)5-s + (−0.172 + 0.172i)7-s + (−0.225 − 0.974i)8-s + (0.784 − 0.164i)10-s + (0.841 + 0.348i)11-s + (−0.359 − 0.867i)13-s + (−0.0451 + 0.239i)14-s + (−0.735 − 0.677i)16-s + 0.862·17-s + (−1.28 + 0.533i)19-s + (0.555 − 0.578i)20-s + (0.891 − 0.186i)22-s + (−0.855 + 0.855i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.618 + 0.785i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94700 - 0.945292i\)
\(L(\frac12)\) \(\approx\) \(1.94700 - 0.945292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 0.797i)T \)
3 \( 1 \)
good5 \( 1 + (-1.65 - 0.686i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.456 - 0.456i)T - 7iT^{2} \)
11 \( 1 + (-2.79 - 1.15i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.29 + 3.12i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
19 \( 1 + (5.61 - 2.32i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.10 - 4.10i)T - 23iT^{2} \)
29 \( 1 + (-1.87 - 4.53i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 0.580iT - 31T^{2} \)
37 \( 1 + (-2.58 + 6.22i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.98 - 2.98i)T + 41iT^{2} \)
43 \( 1 + (2.78 - 6.72i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 8.67iT - 47T^{2} \)
53 \( 1 + (3.70 - 8.95i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.77 - 4.28i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (9.49 - 3.93i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-3.50 - 8.46i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (7.84 + 7.84i)T + 71iT^{2} \)
73 \( 1 + (-10.7 + 10.7i)T - 73iT^{2} \)
79 \( 1 + 5.27T + 79T^{2} \)
83 \( 1 + (1.80 + 4.36i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-12.4 + 12.4i)T - 89iT^{2} \)
97 \( 1 - 9.99T + 97T^{2} \)
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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

−11.93779443108207415121100515201, −10.66826748539141772948908217741, −10.05695117213882711828480886813, −9.206609307542409519372469423497, −7.63040977313057547255574394244, −6.29712270101525563793750315114, −5.72064778428123796516294200118, −4.34604615671232326032364173125, −3.08603583731486300372912383446, −1.74923852345284406621755950924, 2.17409069145718203230768364834, 3.81367880652416590854703582575, 4.83008202937696521281441930846, 6.11520802526020115646750157320, 6.65690928963515562165823429330, 8.003528489167554546593592711239, 8.988202765250967928347155398005, 9.981020981071636437697257786556, 11.29893084714009026281945059315, 12.14803072890073488632589238828

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