Properties

Label 2-288-32.13-c1-0-10
Degree $2$
Conductor $288$
Sign $-0.628 + 0.777i$
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  + (−0.716 − 1.21i)2-s + (−0.971 + 1.74i)4-s + (−2.42 + 1.00i)5-s + (3.40 − 3.40i)7-s + (2.82 − 0.0685i)8-s + (2.96 + 2.23i)10-s + (−0.847 − 2.04i)11-s + (−1.82 − 0.757i)13-s + (−6.59 − 1.71i)14-s + (−2.11 − 3.39i)16-s − 7.04i·17-s + (−4.96 − 2.05i)19-s + (0.600 − 5.20i)20-s + (−1.88 + 2.49i)22-s + (3.37 + 3.37i)23-s + ⋯
L(s)  = 1  + (−0.506 − 0.861i)2-s + (−0.485 + 0.873i)4-s + (−1.08 + 0.448i)5-s + (1.28 − 1.28i)7-s + (0.999 − 0.0242i)8-s + (0.936 + 0.706i)10-s + (−0.255 − 0.616i)11-s + (−0.507 − 0.210i)13-s + (−1.76 − 0.457i)14-s + (−0.527 − 0.849i)16-s − 1.70i·17-s + (−1.14 − 0.472i)19-s + (0.134 − 1.16i)20-s + (−0.402 + 0.532i)22-s + (0.703 + 0.703i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.628 + 0.777i$
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: \((2,\ 288,\ (\ :1/2),\ -0.628 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318458 - 0.666704i\)
\(L(\frac12)\) \(\approx\) \(0.318458 - 0.666704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.716 + 1.21i)T \)
3 \( 1 \)
good5 \( 1 + (2.42 - 1.00i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.40 + 3.40i)T - 7iT^{2} \)
11 \( 1 + (0.847 + 2.04i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.82 + 0.757i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 7.04iT - 17T^{2} \)
19 \( 1 + (4.96 + 2.05i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.37 - 3.37i)T + 23iT^{2} \)
29 \( 1 + (-2.83 + 6.84i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.94T + 31T^{2} \)
37 \( 1 + (-4.02 + 1.66i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.970 + 0.970i)T + 41iT^{2} \)
43 \( 1 + (0.0467 + 0.112i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 6.49iT - 47T^{2} \)
53 \( 1 + (-0.894 - 2.15i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-7.71 + 3.19i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (4.20 - 10.1i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-0.933 + 2.25i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (4.81 - 4.81i)T - 71iT^{2} \)
73 \( 1 + (2.31 + 2.31i)T + 73iT^{2} \)
79 \( 1 - 6.41iT - 79T^{2} \)
83 \( 1 + (-6.76 - 2.80i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.37 - 8.37i)T - 89iT^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
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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

−11.26161445596509026820379173817, −10.88637386339582055954175853426, −9.807162971987492243960202332382, −8.497901751585688357751436958171, −7.66698038538686077247508278068, −7.17640901459981361363148403224, −4.83346191710930761142445227997, −4.05532445220397663417607870334, −2.72851550887195773507757715093, −0.68570091726959870522212072537, 1.87399848837771422782733881256, 4.35665224829517162943381481953, 5.05244129507234336757399093024, 6.28867771097359912555698439792, 7.58512810910733011521685952353, 8.529277447418185011923096086507, 8.615629699173877427031161960053, 10.21475970523188151499151867600, 11.14108782996867839048919201894, 12.20589912408144221072805270482

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