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Properties

Label 2-288-32.29-c1-0-0
Degree $2$
Conductor $288$
Sign $-0.924 + 0.381i$
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  + (0.165 + 1.40i)2-s + (−1.94 + 0.465i)4-s + (−0.805 − 1.94i)5-s + (−2.62 + 2.62i)7-s + (−0.976 − 2.65i)8-s + (2.59 − 1.45i)10-s + (−5.53 + 2.29i)11-s + (−2.40 + 5.80i)13-s + (−4.12 − 3.25i)14-s + (3.56 − 1.81i)16-s − 4.07i·17-s + (0.296 − 0.716i)19-s + (2.47 + 3.40i)20-s + (−4.13 − 7.38i)22-s + (1.87 + 1.87i)23-s + ⋯
L(s)  = 1  + (0.117 + 0.993i)2-s + (−0.972 + 0.232i)4-s + (−0.360 − 0.869i)5-s + (−0.991 + 0.991i)7-s + (−0.345 − 0.938i)8-s + (0.821 − 0.459i)10-s + (−1.66 + 0.690i)11-s + (−0.667 + 1.61i)13-s + (−1.10 − 0.868i)14-s + (0.891 − 0.452i)16-s − 0.989i·17-s + (0.0681 − 0.164i)19-s + (0.552 + 0.761i)20-s + (−0.881 − 1.57i)22-s + (0.390 + 0.390i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.924 + 0.381i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.924 + 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0773086 - 0.390074i\)
\(L(\frac12)\) \(\approx\) \(0.0773086 - 0.390074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.165 - 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (0.805 + 1.94i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.62 - 2.62i)T - 7iT^{2} \)
11 \( 1 + (5.53 - 2.29i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.40 - 5.80i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 4.07iT - 17T^{2} \)
19 \( 1 + (-0.296 + 0.716i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.87 - 1.87i)T + 23iT^{2} \)
29 \( 1 + (-3.86 - 1.60i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 4.49T + 31T^{2} \)
37 \( 1 + (-0.862 - 2.08i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.95 + 5.95i)T + 41iT^{2} \)
43 \( 1 + (6.19 - 2.56i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 3.80iT - 47T^{2} \)
53 \( 1 + (4.67 - 1.93i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.947 - 2.28i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (7.97 + 3.30i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (4.06 + 1.68i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (7.57 - 7.57i)T - 71iT^{2} \)
73 \( 1 + (-6.57 - 6.57i)T + 73iT^{2} \)
79 \( 1 - 6.37iT - 79T^{2} \)
83 \( 1 + (1.17 - 2.83i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-1.74 + 1.74i)T - 89iT^{2} \)
97 \( 1 - 4.50T + 97T^{2} \)
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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

−12.46015244033347026390780798992, −11.85743507426515633982293348204, −9.997982088322482144558317362319, −9.301078729713465247690577476160, −8.527038528934045112472417807632, −7.41854337342116428656595694127, −6.52214193897901528583834676351, −5.16401665700470237156192981459, −4.64362044290294463566536551693, −2.80155466767962296941376752702, 0.27357809132646673998715428021, 2.89979775403108334921299767236, 3.36250113279554179441874463808, 4.93710373143920880999647561060, 6.18929557124564875513191513160, 7.58796359572640690261518370654, 8.350831721884642801360731939903, 10.11170544426911878623506820673, 10.32174455817613725232062615603, 10.96083216991282843301268288726

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