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Properties

Label 2-288-96.59-c1-0-15
Degree $2$
Conductor $288$
Sign $-0.224 + 0.974i$
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  + (1.04 − 0.948i)2-s + (0.201 − 1.98i)4-s + (1.39 − 3.36i)5-s + (−1.05 + 1.05i)7-s + (−1.67 − 2.27i)8-s + (−1.73 − 4.85i)10-s + (−1.50 + 3.64i)11-s + (−2.23 + 0.927i)13-s + (−0.106 + 2.11i)14-s + (−3.91 − 0.803i)16-s + 7.49·17-s + (0.818 + 1.97i)19-s + (−6.42 − 3.45i)20-s + (1.87 + 5.25i)22-s + (5.80 − 5.80i)23-s + ⋯
L(s)  = 1  + (0.741 − 0.670i)2-s + (0.100 − 0.994i)4-s + (0.624 − 1.50i)5-s + (−0.399 + 0.399i)7-s + (−0.592 − 0.805i)8-s + (−0.547 − 1.53i)10-s + (−0.455 + 1.09i)11-s + (−0.620 + 0.257i)13-s + (−0.0285 + 0.564i)14-s + (−0.979 − 0.200i)16-s + 1.81·17-s + (0.187 + 0.453i)19-s + (−1.43 − 0.773i)20-s + (0.399 + 1.12i)22-s + (1.21 − 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.224 + 0.974i$
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: \((2,\ 288,\ (\ :1/2),\ -0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19106 - 1.49659i\)
\(L(\frac12)\) \(\approx\) \(1.19106 - 1.49659i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 + 0.948i)T \)
3 \( 1 \)
good5 \( 1 + (-1.39 + 3.36i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.05 - 1.05i)T - 7iT^{2} \)
11 \( 1 + (1.50 - 3.64i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.23 - 0.927i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 7.49T + 17T^{2} \)
19 \( 1 + (-0.818 - 1.97i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.80 + 5.80i)T - 23iT^{2} \)
29 \( 1 + (0.326 - 0.135i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.71iT - 31T^{2} \)
37 \( 1 + (-0.387 - 0.160i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.50 - 1.50i)T + 41iT^{2} \)
43 \( 1 + (-7.40 - 3.06i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 7.27iT - 47T^{2} \)
53 \( 1 + (3.94 + 1.63i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (12.7 + 5.30i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.579 + 1.40i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (7.96 - 3.29i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (4.75 + 4.75i)T + 71iT^{2} \)
73 \( 1 + (7.99 - 7.99i)T - 73iT^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (1.35 - 0.560i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.75 + 4.75i)T - 89iT^{2} \)
97 \( 1 + 1.28T + 97T^{2} \)
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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

−12.05843394587826102442549183747, −10.53037575898101676587862757615, −9.645829280228284534702828814863, −9.161051396177110023053596779701, −7.67240052027523606508288725276, −6.14132669396933427933985531129, −5.17866411966229962437868757242, −4.53797973329973152379896609991, −2.79260687453899908960236729392, −1.35063564988639649901028759499, 2.87272533525373124301207978604, 3.44309579463704637012814315904, 5.35538048344359973259167116780, 6.04855015720449351136572102955, 7.18202006476356336137317570179, 7.69758774772426776938013189011, 9.273777817368261237353256493279, 10.35474779305683188090397442332, 11.13228473629178043971154799216, 12.18800927081291482862191694725

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