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Properties

Label 1-1027-1027.1026-r1-0-0
Degree $1$
Conductor $1027$
Sign $1$
Analytic cond. $110.366$
Root an. cond. $110.366$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s − 27-s + 28-s − 29-s − 30-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s + 24-s + 25-s − 27-s + 28-s − 29-s − 30-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $1$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1027} (1026, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3525096550\)
\(L(\frac12)\) \(\approx\) \(0.3525096550\)
\(L(1)\) \(\approx\) \(0.3921250997\)
\(L(1)\) \(\approx\) \(0.3921250997\)

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 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−21.21617524816555922407620284692, −20.58875935332382012046319065897, −19.71541902612029971121659059724, −18.7174923703805286420378412055, −18.34490716242068656163985078151, −17.504356957589441045491495364905, −16.81672193460605829175709747431, −16.085799186469202153799160041987, −15.2506359530541952828436592534, −14.86764313698575037324858355041, −13.0801933267665083776751023403, −12.43982613825653677212348529939, −11.32938055071072917414550692636, −11.088580693114777569491834142494, −10.50906740336628070278215425904, −9.22037926833753809162212894950, −8.34736709219318274025347966242, −7.55189854572797291885309960730, −6.994558120206592778655915246706, −5.86313607969680177805654018278, −4.9124650186722267699944259050, −4.04344300003021797662969416595, −2.571023258739202350473125776014, −1.49726339639022017040936701068, −0.346697042022509007371337704903, 0.346697042022509007371337704903, 1.49726339639022017040936701068, 2.571023258739202350473125776014, 4.04344300003021797662969416595, 4.9124650186722267699944259050, 5.86313607969680177805654018278, 6.994558120206592778655915246706, 7.55189854572797291885309960730, 8.34736709219318274025347966242, 9.22037926833753809162212894950, 10.50906740336628070278215425904, 11.088580693114777569491834142494, 11.32938055071072917414550692636, 12.43982613825653677212348529939, 13.0801933267665083776751023403, 14.86764313698575037324858355041, 15.2506359530541952828436592534, 16.085799186469202153799160041987, 16.81672193460605829175709747431, 17.504356957589441045491495364905, 18.34490716242068656163985078151, 18.7174923703805286420378412055, 19.71541902612029971121659059724, 20.58875935332382012046319065897, 21.21617524816555922407620284692

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