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Properties

Label 2-1288-1288.1189-c0-0-1
Degree $2$
Conductor $1288$
Sign $-0.270 - 0.962i$
Analytic cond. $0.642795$
Root an. cond. $0.801745$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (−0.415 + 0.909i)9-s + (−0.817 + 0.708i)11-s + (0.841 + 0.540i)14-s + (−0.959 − 0.281i)16-s + (−0.959 + 0.281i)18-s + (−1.07 − 0.153i)22-s + (0.415 + 0.909i)23-s + (0.654 − 0.755i)25-s + (0.142 + 0.989i)28-s + (1.49 + 0.215i)29-s + (−0.415 − 0.909i)32-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (−0.415 + 0.909i)9-s + (−0.817 + 0.708i)11-s + (0.841 + 0.540i)14-s + (−0.959 − 0.281i)16-s + (−0.959 + 0.281i)18-s + (−1.07 − 0.153i)22-s + (0.415 + 0.909i)23-s + (0.654 − 0.755i)25-s + (0.142 + 0.989i)28-s + (1.49 + 0.215i)29-s + (−0.415 − 0.909i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $-0.270 - 0.962i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ -0.270 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.460961615\)
\(L(\frac12)\) \(\approx\) \(1.460961615\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
good3 \( 1 + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.654 + 0.755i)T^{2} \)
11 \( 1 + (0.817 - 0.708i)T + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (-1.49 - 0.215i)T + (0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.512 + 0.234i)T + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (0.983 + 1.53i)T + (-0.415 + 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (1.49 + 1.29i)T + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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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

−10.29158035157020768386010791389, −8.917141778538024414277191140638, −8.254555619991238153294169100381, −7.58256906951484870928681041851, −6.95894980993412772682374386256, −5.74744442367250492967271309208, −4.94663256409233600270204780909, −4.53756074477589192038791130404, −3.13306270826854243640196023132, −2.06989982354470952155384719500, 1.10511565493187661066442799508, 2.57329590643575108802434121288, 3.29939206678765687445343778385, 4.53615753650304592545942639643, 5.23213293574460777776795869211, 6.06468253649586397966999265251, 6.94221075645050418876759412864, 8.375901835732327923689223706936, 8.714046498514682020354844191684, 9.838585806163045499098017201755

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