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Properties

Label 1-1027-1027.971-r0-0-0
Degree $1$
Conductor $1027$
Sign $0.985 - 0.168i$
Analytic cond. $4.76936$
Root an. cond. $4.76936$
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  + 2-s + 3-s + 4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + 12-s + 14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + 12-s + 14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(4.76936\)
Root analytic conductor: \(4.76936\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (0:\ ),\ 0.985 - 0.168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.351078532 - 0.3692711755i\)
\(L(\frac12)\) \(\approx\) \(4.351078532 - 0.3692711755i\)
\(L(1)\) \(\approx\) \(2.720180851 - 0.1737608311i\)
\(L(1)\) \(\approx\) \(2.720180851 - 0.1737608311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−21.66802789085271903387865206186, −20.82846740913910822127978750454, −20.224341893221660550005331548554, −19.52915299739436408362063665828, −18.50055241587158365786813920043, −18.09651608937611643243031115012, −16.465221933893756542623757657718, −15.81061824439374206543537347810, −15.09333785195448167085852391403, −14.38686602474118819829969774773, −13.91629666669179907440507447625, −13.20435929138147218213225436756, −12.04597532231472623086673623278, −11.284651381028106676337180081358, −10.70176662673638160057447953142, −9.63201494378859078609870305257, −8.267939796796438387242603149124, −7.77050017064759060253870592877, −7.02219400193662169322293366342, −5.99376233113010389209607592579, −4.81013848235874496358772590911, −4.123544952047005097665636930550, −3.00696295319032787441384550474, −2.66287248785952499438547600488, −1.42398525324602620865352005950, 1.56508260212883036666346305697, 1.99832417953068843518688816605, 3.37137210859593263915140022817, 4.128703294236184621031202350725, 4.84655722185946168652461983996, 5.59800392702703274892968611120, 7.15846675555713797225427332369, 7.688590396903628498462321369663, 8.37755195257501737011377981816, 9.42431676292842653694181971963, 10.44070535923920344124306054633, 11.470160365674609721246461766544, 12.20099460805020232011028695999, 13.039247187963107362503282942238, 13.54827107866701114054903907527, 14.55818796643961154089802591895, 15.16012453964026199598545200070, 15.68525286642532644587217827030, 16.60027511613864668367614739219, 17.62045663867273011110443537361, 18.60405098998032043894328463171, 19.7481908318020143513309469702, 20.24111918903093265274763035640, 20.633430647319669360363198298293, 21.46401743635481538568335985062

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