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Properties

Label 1-1288-1288.187-r0-0-0
Degree $1$
Conductor $1288$
Sign $0.986 - 0.162i$
Analytic cond. $5.98144$
Root an. cond. $5.98144$
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  + (−0.981 + 0.189i)3-s + (−0.786 + 0.618i)5-s + (0.928 − 0.371i)9-s + (0.723 − 0.690i)11-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.888 − 0.458i)17-s + (0.888 + 0.458i)19-s + (0.235 − 0.971i)25-s + (−0.841 + 0.540i)27-s + (−0.841 − 0.540i)29-s + (−0.327 − 0.945i)31-s + (−0.580 + 0.814i)33-s + (−0.928 + 0.371i)37-s + (−0.580 − 0.814i)39-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)3-s + (−0.786 + 0.618i)5-s + (0.928 − 0.371i)9-s + (0.723 − 0.690i)11-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.888 − 0.458i)17-s + (0.888 + 0.458i)19-s + (0.235 − 0.971i)25-s + (−0.841 + 0.540i)27-s + (−0.841 − 0.540i)29-s + (−0.327 − 0.945i)31-s + (−0.580 + 0.814i)33-s + (−0.928 + 0.371i)37-s + (−0.580 − 0.814i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $0.986 - 0.162i$
Analytic conductor: \(5.98144\)
Root analytic conductor: \(5.98144\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1288,\ (0:\ ),\ 0.986 - 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9446816376 - 0.07741295469i\)
\(L(\frac12)\) \(\approx\) \(0.9446816376 - 0.07741295469i\)
\(L(1)\) \(\approx\) \(0.7695973289 + 0.05491475343i\)
\(L(1)\) \(\approx\) \(0.7695973289 + 0.05491475343i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.981 + 0.189i)T \)
5 \( 1 + (-0.786 + 0.618i)T \)
11 \( 1 + (0.723 - 0.690i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.888 - 0.458i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.928 + 0.371i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.580 - 0.814i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (0.981 + 0.189i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.0475 - 0.998i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.93235121094297284179211652605, −20.264046457746992088281932663581, −19.48900362255037787732682122315, −18.73303145911039382594706883845, −17.77754508683176058785580975155, −17.30440081002438117447936756595, −16.39348851180935057132006772397, −15.86731270612225696600710308054, −15.08919915525639137168232939752, −14.10980743609968794466381607753, −12.80774112985802955980018196306, −12.606569072145178886829242995131, −11.73274312284897688725298922873, −11.091023702687109077790759183759, −10.18952004371932917235483412910, −9.31362311184631809333508300713, −8.29112402994513368469773475560, −7.43701611486984691475515330218, −6.82234486241040319717523473287, −5.58385496408397725080464117431, −5.1320371218185008724526142366, −4.08235415941804661447063494966, −3.32132602811911561427974839493, −1.60457404824744497155057398356, −0.89140255066465035553193518703, 0.61345321926079087187707850916, 1.786266424252891077952919270908, 3.44530273611536313515662800643, 3.79560046871632020041280323031, 4.92611334996330712952919298075, 5.85953709444324383626259545708, 6.626094976233267810454310045072, 7.331530187290825557367651030553, 8.26282887784602077517958930012, 9.42136759563234213286215630317, 10.10367286249279157499329630594, 11.1560259370734810080252572447, 11.63365429766165318261193471418, 12.02209702314788601986338607919, 13.256518736901571053454017802268, 14.16768156159943677780236045817, 14.8970770612962013611529279367, 15.82502494901455958365683799532, 16.46921346435540733411389678126, 16.92102897478963048488777777215, 18.09881506212429556840410594068, 18.72908927893627038911198080957, 19.13758245245301040870332831945, 20.294934053256282391141247838002, 21.112299230953967858340712682209

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