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Properties

Label 2-288-288.11-c1-0-15
Degree $2$
Conductor $288$
Sign $0.373 + 0.927i$
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  + (−0.747 − 1.20i)2-s + (−1.18 − 1.26i)3-s + (−0.883 + 1.79i)4-s + (0.396 + 3.01i)5-s + (−0.633 + 2.36i)6-s + (−0.375 − 1.40i)7-s + (2.81 − 0.279i)8-s + (−0.196 + 2.99i)9-s + (3.32 − 2.72i)10-s + (4.40 − 3.38i)11-s + (3.31 − 1.00i)12-s + (2.04 − 2.65i)13-s + (−1.40 + 1.49i)14-s + (3.34 − 4.07i)15-s + (−2.43 − 3.17i)16-s + 3.35·17-s + ⋯
L(s)  = 1  + (−0.528 − 0.849i)2-s + (−0.683 − 0.729i)3-s + (−0.441 + 0.897i)4-s + (0.177 + 1.34i)5-s + (−0.258 + 0.966i)6-s + (−0.142 − 0.530i)7-s + (0.995 − 0.0989i)8-s + (−0.0654 + 0.997i)9-s + (1.05 − 0.863i)10-s + (1.32 − 1.01i)11-s + (0.956 − 0.290i)12-s + (0.565 − 0.737i)13-s + (−0.375 + 0.400i)14-s + (0.862 − 1.05i)15-s + (−0.609 − 0.792i)16-s + 0.814·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.373 + 0.927i$
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: \((2,\ 288,\ (\ :1/2),\ 0.373 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694229 - 0.468600i\)
\(L(\frac12)\) \(\approx\) \(0.694229 - 0.468600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.747 + 1.20i)T \)
3 \( 1 + (1.18 + 1.26i)T \)
good5 \( 1 + (-0.396 - 3.01i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (0.375 + 1.40i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-4.40 + 3.38i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (-2.04 + 2.65i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 - 3.35T + 17T^{2} \)
19 \( 1 + (6.96 - 2.88i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-6.39 - 1.71i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.25 - 0.165i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (-6.64 + 3.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.32 + 3.20i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.0728 - 0.271i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.59 + 2.07i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (-6.45 - 3.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.64 + 3.95i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (10.2 - 1.35i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (1.30 - 9.89i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (6.34 - 8.26i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (2.27 + 2.27i)T + 71iT^{2} \)
73 \( 1 + (-1.62 + 1.62i)T - 73iT^{2} \)
79 \( 1 + (-2.99 + 5.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.484 - 0.0637i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (8.26 - 8.26i)T - 89iT^{2} \)
97 \( 1 + (3.50 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.41705425358115292257212574395, −10.73443957585275122846214646155, −10.26632628987085451518535502177, −8.792202184819705225727761997929, −7.74352358141965886214253658159, −6.76014989587122604931276116573, −5.95558556859461050207849006678, −3.94758149744094317177518773973, −2.82226775069112610430094181467, −1.09841682960352054991591181840, 1.24723157688814332283424935516, 4.35884370354565829993386003609, 4.85576146958525580569901078942, 6.11174486154851882224024774190, 6.81906180516562663251524857766, 8.625506095824324519757969715298, 9.033687481124335200458157175800, 9.741738144474616101416073575781, 10.87037180894450574548657752397, 12.04946777776921961820345490361

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