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Properties

Label 2-1288-1288.13-c0-0-1
Degree $2$
Conductor $1288$
Sign $-0.0117 - 0.999i$
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 + (−0.909 − 0.584i)3-s + (−0.142 + 0.989i)4-s + (−0.234 + 0.512i)5-s + (−0.153 − 1.07i)6-s + (0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.0702 + 0.153i)9-s + (−0.540 + 0.158i)10-s + (0.708 − 0.817i)12-s + (1.45 − 0.425i)13-s + (0.415 + 0.909i)14-s + (0.512 − 0.329i)15-s + (−0.959 − 0.281i)16-s + (−0.0702 + 0.153i)18-s + (−0.258 + 1.80i)19-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.909 − 0.584i)3-s + (−0.142 + 0.989i)4-s + (−0.234 + 0.512i)5-s + (−0.153 − 1.07i)6-s + (0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.0702 + 0.153i)9-s + (−0.540 + 0.158i)10-s + (0.708 − 0.817i)12-s + (1.45 − 0.425i)13-s + (0.415 + 0.909i)14-s + (0.512 − 0.329i)15-s + (−0.959 − 0.281i)16-s + (−0.0702 + 0.153i)18-s + (−0.258 + 1.80i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $-0.0117 - 0.999i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ -0.0117 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129640578\)
\(L(\frac12)\) \(\approx\) \(1.129640578\)
\(L(1)\) 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 \( 1 + (0.959 - 0.281i)T \)
good3 \( 1 + (0.909 + 0.584i)T + (0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.234 - 0.512i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (-1.45 + 0.425i)T + (0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.258 - 1.80i)T + (-0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.654 - 0.755i)T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.415 - 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (-1.74 + 0.512i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2} \)
83 \( 1 + (0.822 + 1.80i)T + (-0.654 + 0.755i)T^{2} \)
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

−10.31305981790707644401511859081, −8.858370740854464821792371900487, −8.146148472231723261983421888276, −7.52189599364226851965624318102, −6.55616496535629804509190760779, −5.87528644451259814257931293785, −5.42506344050251372805174145499, −4.13197589693405225729346308720, −3.31732204903245224049524422014, −1.65307003948230781842473856229, 0.982137622351988802136592780138, 2.37589550201853217347796073456, 3.95637830218696712839637057495, 4.48044828000001067390389585693, 5.16693648677764396679078488162, 5.98656950225212895351403013862, 6.89192680191426720449476862204, 8.351635975693483022672933716593, 8.884465546672846262519641519636, 10.03356886160327266202232611987

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