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Properties

Label 1-1077-1077.1076-r0-0-0
Degree $1$
Conductor $1077$
Sign $1$
Analytic cond. $5.00156$
Root an. cond. $5.00156$
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 + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 20-s + 22-s − 23-s + 25-s + 26-s − 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s + 38-s + 40-s − 41-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 20-s + 22-s − 23-s + 25-s + 26-s − 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s + 38-s + 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1077\)    =    \(3 \cdot 359\)
Sign: $1$
Analytic conductor: \(5.00156\)
Root analytic conductor: \(5.00156\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1077} (1076, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1077,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2004290220\)
\(L(\frac12)\) \(\approx\) \(0.2004290220\)
\(L(1)\) \(\approx\) \(0.3588840292\)
\(L(1)\) \(\approx\) \(0.3588840292\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
359 \( 1 \)
good2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
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 \( 1 + T \)
79 \( 1 + T \)
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

−21.42999349751086781947677325753, −20.16810906187711536095561190028, −19.8582502581599908728506539793, −19.205586007248427294888497209801, −18.421145226022566405880602585607, −17.72543579151765653389986978226, −16.53805014351104470044857657045, −16.279110795549850425478704807610, −15.27906223031700732739140864686, −14.95275397447666301697730097952, −13.395274008835531455974759989969, −12.526469953849344657818473539, −11.93957856079874813853487473210, −10.915088982901455292710466454672, −10.280175482126348942711881732461, −9.47984876076631061976845039625, −8.49041224967925116799757098518, −7.88726798856948207827732023923, −6.96498248009501821680668107628, −6.376630881782770969576255210224, −5.07096948656643262872215009228, −3.91347832047517333201165539650, −2.87565825991942696093892489352, −2.13040612448085185381372808667, −0.33754087979929111989857674622, 0.33754087979929111989857674622, 2.13040612448085185381372808667, 2.87565825991942696093892489352, 3.91347832047517333201165539650, 5.07096948656643262872215009228, 6.376630881782770969576255210224, 6.96498248009501821680668107628, 7.88726798856948207827732023923, 8.49041224967925116799757098518, 9.47984876076631061976845039625, 10.280175482126348942711881732461, 10.915088982901455292710466454672, 11.93957856079874813853487473210, 12.526469953849344657818473539, 13.395274008835531455974759989969, 14.95275397447666301697730097952, 15.27906223031700732739140864686, 16.279110795549850425478704807610, 16.53805014351104470044857657045, 17.72543579151765653389986978226, 18.421145226022566405880602585607, 19.205586007248427294888497209801, 19.8582502581599908728506539793, 20.16810906187711536095561190028, 21.42999349751086781947677325753

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