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Properties

Label 2-1288-1288.1133-c0-0-4
Degree $2$
Conductor $1288$
Sign $0.854 + 0.519i$
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  + (−0.415 + 0.909i)2-s + (−0.540 − 0.158i)3-s + (−0.654 − 0.755i)4-s + (1.66 − 1.07i)5-s + (0.368 − 0.425i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (−0.574 − 0.368i)9-s + (0.281 + 1.95i)10-s + (0.234 + 0.512i)12-s + (0.258 + 1.80i)13-s + (0.841 + 0.540i)14-s + (−1.07 + 0.314i)15-s + (−0.142 + 0.989i)16-s + (0.574 − 0.368i)18-s + (−0.708 − 0.817i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−0.540 − 0.158i)3-s + (−0.654 − 0.755i)4-s + (1.66 − 1.07i)5-s + (0.368 − 0.425i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (−0.574 − 0.368i)9-s + (0.281 + 1.95i)10-s + (0.234 + 0.512i)12-s + (0.258 + 1.80i)13-s + (0.841 + 0.540i)14-s + (−1.07 + 0.314i)15-s + (−0.142 + 0.989i)16-s + (0.574 − 0.368i)18-s + (−0.708 − 0.817i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ 0.854 + 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8560269284\)
\(L(\frac12)\) \(\approx\) \(0.8560269284\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
good3 \( 1 + (0.540 + 0.158i)T + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (-1.66 + 1.07i)T + (0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (-0.258 - 1.80i)T + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.708 + 0.817i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.153 - 1.07i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.345 - 0.755i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (-1.27 - 0.817i)T + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−9.543332153687592608315267157445, −8.936068322118238281317741489771, −8.448405176519454587918403508735, −6.98420958786216254968204690998, −6.46475140736504712172971755104, −5.86309559206926036691520978540, −4.84342369020199724224268700169, −4.32458965292457510164904722648, −2.03886146432110274231945961839, −0.952729328405093493901870066743, 1.77584620558823344072405411188, 2.63887561394703417737344141768, 3.34486885923734321458130592775, 5.17990134234269268743818590720, 5.63665660956966740784301197375, 6.34173715908616104818104452046, 7.72833470982933981216642832348, 8.500265181957457057456427940690, 9.436504246440060921550036928747, 10.09819211736575965193262534507

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