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Properties

Label 2-288-96.59-c1-0-12
Degree $2$
Conductor $288$
Sign $-0.652 + 0.757i$
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  + (−0.450 − 1.34i)2-s + (−1.59 + 1.20i)4-s + (0.739 − 1.78i)5-s + (0.385 − 0.385i)7-s + (2.33 + 1.59i)8-s + (−2.72 − 0.188i)10-s + (2.36 − 5.70i)11-s + (−2.30 + 0.956i)13-s + (−0.690 − 0.343i)14-s + (1.08 − 3.84i)16-s − 5.05·17-s + (−1.27 − 3.07i)19-s + (0.975 + 3.74i)20-s + (−8.71 − 0.601i)22-s + (2.28 − 2.28i)23-s + ⋯
L(s)  = 1  + (−0.318 − 0.948i)2-s + (−0.797 + 0.603i)4-s + (0.330 − 0.798i)5-s + (0.145 − 0.145i)7-s + (0.825 + 0.564i)8-s + (−0.862 − 0.0594i)10-s + (0.712 − 1.72i)11-s + (−0.640 + 0.265i)13-s + (−0.184 − 0.0917i)14-s + (0.271 − 0.962i)16-s − 1.22·17-s + (−0.292 − 0.705i)19-s + (0.218 + 0.836i)20-s + (−1.85 − 0.128i)22-s + (0.476 − 0.476i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.652 + 0.757i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419581 - 0.914631i\)
\(L(\frac12)\) \(\approx\) \(0.419581 - 0.914631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.450 + 1.34i)T \)
3 \( 1 \)
good5 \( 1 + (-0.739 + 1.78i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.385 + 0.385i)T - 7iT^{2} \)
11 \( 1 + (-2.36 + 5.70i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.30 - 0.956i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 + (1.27 + 3.07i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.28 + 2.28i)T - 23iT^{2} \)
29 \( 1 + (0.735 - 0.304i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 3.40iT - 31T^{2} \)
37 \( 1 + (-9.56 - 3.96i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.27 + 5.27i)T + 41iT^{2} \)
43 \( 1 + (-2.53 - 1.05i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 6.85iT - 47T^{2} \)
53 \( 1 + (-7.45 - 3.08i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (6.14 + 2.54i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.67 - 6.46i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-10.2 + 4.26i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-6.37 - 6.37i)T + 71iT^{2} \)
73 \( 1 + (-9.03 + 9.03i)T - 73iT^{2} \)
79 \( 1 - 1.22T + 79T^{2} \)
83 \( 1 + (-14.8 + 6.14i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.97 - 4.97i)T - 89iT^{2} \)
97 \( 1 - 1.23T + 97T^{2} \)
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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

−11.29047354391625814134221388044, −10.81321743655338627703254607069, −9.308407916441656047168541506059, −8.998065037693440641856957762447, −8.011193984418884545641237043963, −6.49328646745601819394393952791, −5.06926542528572728768473265232, −4.06830638232353144801075484393, −2.57527373460272786780631144006, −0.892978976657856062852026102640, 2.09112335297794020528390698898, 4.15585576271241329137927456480, 5.22730220965448799086725839883, 6.62437565548104265441791104649, 7.02580193346455012840512216133, 8.190795745618734520060318904404, 9.402941101000858387263662611063, 9.975681771140457520479354037417, 10.96783342168172848139852934181, 12.27175198812869355489780830264

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