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Properties

Label 2-288-288.11-c1-0-27
Degree $2$
Conductor $288$
Sign $0.908 + 0.418i$
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  + (0.808 − 1.16i)2-s + (1.35 + 1.07i)3-s + (−0.691 − 1.87i)4-s + (0.174 + 1.32i)5-s + (2.34 − 0.709i)6-s + (0.471 + 1.75i)7-s + (−2.73 − 0.715i)8-s + (0.696 + 2.91i)9-s + (1.67 + 0.869i)10-s + (4.41 − 3.38i)11-s + (1.07 − 3.29i)12-s + (2.31 − 3.01i)13-s + (2.42 + 0.875i)14-s + (−1.18 + 1.98i)15-s + (−3.04 + 2.59i)16-s − 2.05·17-s + ⋯
L(s)  = 1  + (0.571 − 0.820i)2-s + (0.784 + 0.619i)3-s + (−0.345 − 0.938i)4-s + (0.0780 + 0.592i)5-s + (0.957 − 0.289i)6-s + (0.178 + 0.664i)7-s + (−0.967 − 0.252i)8-s + (0.232 + 0.972i)9-s + (0.531 + 0.275i)10-s + (1.33 − 1.02i)11-s + (0.309 − 0.950i)12-s + (0.642 − 0.837i)13-s + (0.647 + 0.234i)14-s + (−0.306 + 0.513i)15-s + (−0.760 + 0.649i)16-s − 0.498·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.908 + 0.418i$
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: \((2,\ 288,\ (\ :1/2),\ 0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08476 - 0.457131i\)
\(L(\frac12)\) \(\approx\) \(2.08476 - 0.457131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.808 + 1.16i)T \)
3 \( 1 + (-1.35 - 1.07i)T \)
good5 \( 1 + (-0.174 - 1.32i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-0.471 - 1.75i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-4.41 + 3.38i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (-2.31 + 3.01i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 2.05T + 17T^{2} \)
19 \( 1 + (4.68 - 1.93i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.94 + 1.59i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.89 + 0.644i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (2.40 - 1.38i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.31 - 8.01i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.79 + 10.4i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.11 - 2.76i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (8.41 + 4.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.13 + 5.15i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.41 - 0.317i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (0.513 - 3.89i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (3.67 - 4.79i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-6.25 - 6.25i)T + 71iT^{2} \)
73 \( 1 + (-8.65 + 8.65i)T - 73iT^{2} \)
79 \( 1 + (4.84 - 8.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.49 + 0.460i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (-3.07 + 3.07i)T - 89iT^{2} \)
97 \( 1 + (-2.99 + 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.59076479738943700826281337436, −10.81663855982188148988703157105, −10.10007897257737904973527242838, −8.903060804215797872380201276105, −8.432512522144015603371319749420, −6.50218491684940967391146997111, −5.56295136722589133643908808208, −4.07761098484591820379316618898, −3.30980932252584010443237464516, −2.03669456846823450465420986966, 1.83746384936892345413012951382, 3.86936296802705987316123338910, 4.43864408614839923352304831500, 6.23968356775891105224749180196, 6.93882373007972198160605949054, 7.84448743495543697976193621759, 8.959061609692538076703566905066, 9.376202431207052267933894181521, 11.25639932352581567696297349830, 12.31407370921339404796806678948

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