Properties

Label 2-288-288.275-c1-0-9
Degree $2$
Conductor $288$
Sign $-0.472 - 0.881i$
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  + (−1.35 − 0.413i)2-s + (0.986 + 1.42i)3-s + (1.65 + 1.11i)4-s + (−1.47 − 0.194i)5-s + (−0.745 − 2.33i)6-s + (−0.772 + 2.88i)7-s + (−1.78 − 2.19i)8-s + (−1.05 + 2.80i)9-s + (1.91 + 0.872i)10-s + (0.930 − 1.21i)11-s + (0.0433 + 3.46i)12-s + (−4.09 + 3.14i)13-s + (2.23 − 3.58i)14-s + (−1.17 − 2.29i)15-s + (1.49 + 3.70i)16-s + 0.887·17-s + ⋯
L(s)  = 1  + (−0.956 − 0.292i)2-s + (0.569 + 0.821i)3-s + (0.829 + 0.559i)4-s + (−0.659 − 0.0868i)5-s + (−0.304 − 0.952i)6-s + (−0.292 + 1.09i)7-s + (−0.629 − 0.777i)8-s + (−0.351 + 0.936i)9-s + (0.605 + 0.275i)10-s + (0.280 − 0.365i)11-s + (0.0125 + 0.999i)12-s + (−1.13 + 0.872i)13-s + (0.598 − 0.957i)14-s + (−0.304 − 0.591i)15-s + (0.374 + 0.927i)16-s + 0.215·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.472 - 0.881i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.472 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351588 + 0.587273i\)
\(L(\frac12)\) \(\approx\) \(0.351588 + 0.587273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.413i)T \)
3 \( 1 + (-0.986 - 1.42i)T \)
good5 \( 1 + (1.47 + 0.194i)T + (4.82 + 1.29i)T^{2} \)
7 \( 1 + (0.772 - 2.88i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.930 + 1.21i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 + (4.09 - 3.14i)T + (3.36 - 12.5i)T^{2} \)
17 \( 1 - 0.887T + 17T^{2} \)
19 \( 1 + (-1.18 + 2.85i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.57 - 1.22i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.945 - 7.17i)T + (-28.0 + 7.50i)T^{2} \)
31 \( 1 + (-7.16 - 4.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.13 + 2.54i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.37 - 8.84i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.438 - 0.336i)T + (11.1 + 41.5i)T^{2} \)
47 \( 1 + (6.71 - 3.87i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.92 + 0.796i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.77 + 13.5i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-3.56 + 0.468i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (-10.8 + 8.32i)T + (17.3 - 64.7i)T^{2} \)
71 \( 1 + (-4.27 + 4.27i)T - 71iT^{2} \)
73 \( 1 + (-2.09 - 2.09i)T + 73iT^{2} \)
79 \( 1 + (3.53 + 6.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.307 - 2.33i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (-4.22 - 4.22i)T + 89iT^{2} \)
97 \( 1 + (-3.95 - 6.85i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.80461983171505739694075214898, −11.20573465617279657506744829051, −9.862930674031998077709309023271, −9.391416671773768944390363940467, −8.495391808177278735512065813639, −7.72336313045920616292860879175, −6.39671472011811688167532665647, −4.81139808155272196170583321215, −3.42121274513155161823174331925, −2.34540674163465239193436471874, 0.62932627318860108968869523614, 2.45564644907702604481688660688, 3.96131034062418908233701697656, 5.93695577793557857352531193326, 7.05780183387817757552650937954, 7.71055720546623525286516175150, 8.230734733430894169526470833138, 9.775538293200244823984760761213, 10.12398114992826228127788092847, 11.62812341588232696009360994887

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