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Properties

Label 2-288-32.5-c1-0-11
Degree $2$
Conductor $288$
Sign $0.232 + 0.972i$
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  + (−1.34 − 0.426i)2-s + (1.63 + 1.15i)4-s + (1.46 + 0.605i)5-s + (−3.54 − 3.54i)7-s + (−1.71 − 2.24i)8-s + (−1.71 − 1.44i)10-s + (0.471 − 1.13i)11-s + (4.97 − 2.05i)13-s + (3.27 + 6.29i)14-s + (1.35 + 3.76i)16-s − 0.419i·17-s + (−0.721 + 0.298i)19-s + (1.69 + 2.67i)20-s + (−1.12 + 1.33i)22-s + (5.76 − 5.76i)23-s + ⋯
L(s)  = 1  + (−0.953 − 0.301i)2-s + (0.817 + 0.575i)4-s + (0.653 + 0.270i)5-s + (−1.34 − 1.34i)7-s + (−0.606 − 0.795i)8-s + (−0.541 − 0.455i)10-s + (0.142 − 0.342i)11-s + (1.37 − 0.570i)13-s + (0.873 + 1.68i)14-s + (0.337 + 0.941i)16-s − 0.101i·17-s + (−0.165 + 0.0685i)19-s + (0.378 + 0.597i)20-s + (−0.238 + 0.284i)22-s + (1.20 − 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.232 + 0.972i$
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: \((2,\ 288,\ (\ :1/2),\ 0.232 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.638184 - 0.503834i\)
\(L(\frac12)\) \(\approx\) \(0.638184 - 0.503834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.426i)T \)
3 \( 1 \)
good5 \( 1 + (-1.46 - 0.605i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (3.54 + 3.54i)T + 7iT^{2} \)
11 \( 1 + (-0.471 + 1.13i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-4.97 + 2.05i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.419iT - 17T^{2} \)
19 \( 1 + (0.721 - 0.298i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.76 + 5.76i)T - 23iT^{2} \)
29 \( 1 + (1.26 + 3.05i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 0.702T + 31T^{2} \)
37 \( 1 + (-1.86 - 0.773i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.76 - 1.76i)T - 41iT^{2} \)
43 \( 1 + (-1.70 + 4.12i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.64iT - 47T^{2} \)
53 \( 1 + (-0.729 + 1.76i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (9.04 + 3.74i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.0348 - 0.0842i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-1.84 - 4.44i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-4.81 - 4.81i)T + 71iT^{2} \)
73 \( 1 + (-4.70 + 4.70i)T - 73iT^{2} \)
79 \( 1 - 2.83iT - 79T^{2} \)
83 \( 1 + (8.15 - 3.37i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.34 - 5.34i)T + 89iT^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−11.13150837204580227442395025628, −10.55327393742496433728898409160, −9.856198539849562404432522142744, −8.952761985920310857407076023208, −7.82983433209760266341929800859, −6.64164384216388118829924319618, −6.17998021101892452485651631113, −3.88639683115223095210009670532, −2.87509340629724157778425809489, −0.866792097483598723148899836117, 1.76136201450385728576647948109, 3.22069563125704281993561907003, 5.47367607603359752818363375061, 6.14918206544548680927873910686, 7.03101778299426705762835690173, 8.572370623425536031395434490505, 9.223954761278749838016391269375, 9.688677834591508058694016866097, 10.94023438645326456969928393431, 11.87086854653272734601832231636

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