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Properties

Label 2-288-288.11-c1-0-37
Degree $2$
Conductor $288$
Sign $0.755 + 0.655i$
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  + (1.39 − 0.240i)2-s + (0.677 − 1.59i)3-s + (1.88 − 0.669i)4-s + (0.183 + 1.39i)5-s + (0.562 − 2.38i)6-s + (0.489 + 1.82i)7-s + (2.46 − 1.38i)8-s + (−2.08 − 2.16i)9-s + (0.590 + 1.89i)10-s + (−1.62 + 1.24i)11-s + (0.211 − 3.45i)12-s + (1.42 − 1.85i)13-s + (1.11 + 2.42i)14-s + (2.34 + 0.652i)15-s + (3.10 − 2.52i)16-s − 2.43·17-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)2-s + (0.391 − 0.920i)3-s + (0.942 − 0.334i)4-s + (0.0821 + 0.623i)5-s + (0.229 − 0.973i)6-s + (0.184 + 0.689i)7-s + (0.871 − 0.489i)8-s + (−0.693 − 0.720i)9-s + (0.186 + 0.600i)10-s + (−0.489 + 0.375i)11-s + (0.0610 − 0.998i)12-s + (0.394 − 0.514i)13-s + (0.299 + 0.648i)14-s + (0.606 + 0.168i)15-s + (0.776 − 0.630i)16-s − 0.590·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33942 - 0.874064i\)
\(L(\frac12)\) \(\approx\) \(2.33942 - 0.874064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.240i)T \)
3 \( 1 + (-0.677 + 1.59i)T \)
good5 \( 1 + (-0.183 - 1.39i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-0.489 - 1.82i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.62 - 1.24i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (-1.42 + 1.85i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 2.43T + 17T^{2} \)
19 \( 1 + (7.41 - 3.07i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (6.65 + 1.78i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.88 - 0.247i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (-5.12 + 2.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.701 - 1.69i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.28 - 8.50i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.75 - 2.28i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (-8.82 - 5.09i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0950 - 0.229i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.79 - 0.762i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (0.474 - 3.60i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-3.74 + 4.88i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (8.45 + 8.45i)T + 71iT^{2} \)
73 \( 1 + (-7.58 + 7.58i)T - 73iT^{2} \)
79 \( 1 + (-0.353 + 0.612i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.7 + 1.54i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (-0.251 + 0.251i)T - 89iT^{2} \)
97 \( 1 + (-6.37 + 11.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.06037417146747241850349761932, −10.97119366287485370314337704002, −10.14557796670937038933800056785, −8.540505717960122345261241008970, −7.71575908884433942826326173805, −6.43355223574177612407218229878, −6.00209557633898602442177226179, −4.39494283388611120909300203267, −2.88435313127436555704310903342, −2.04292808558000243911622657000, 2.34435055194493328722995280000, 3.92181197929293657530380838790, 4.52456000334468806804277755508, 5.61936378396472058404251434850, 6.85343115143400019759526626937, 8.200108283750913988095424096619, 8.879502891807193110947243564594, 10.41458279560342481276164389924, 10.86306274206085501872520754673, 11.97724067497399887535271712294

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