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Properties

Label 1-1057-1057.1056-r0-0-0
Degree $1$
Conductor $1057$
Sign $1$
Analytic cond. $4.90868$
Root an. cond. $4.90868$
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 + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 29-s − 30-s − 31-s + 32-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 29-s − 30-s − 31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $1$
Analytic conductor: \(4.90868\)
Root analytic conductor: \(4.90868\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1057} (1056, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1057,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.957082479\)
\(L(\frac12)\) \(\approx\) \(3.957082479\)
\(L(1)\) \(\approx\) \(2.504406420\)
\(L(1)\) \(\approx\) \(2.504406420\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
151 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
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

−21.54185004488475970277884980129, −20.66990356837076382369186869840, −19.93607853758708818767905584334, −19.59632647511872379771593557683, −18.77137488389543566080119264290, −17.62324229783696251913081365396, −16.190165416679817978486822692889, −15.99709480223371168987506308358, −14.97935113731995842443238277086, −14.543738810554294751857896436288, −13.6958833035159007940730151246, −12.89497858438243603385295537425, −12.21838555940824169094221333961, −11.248394767765015672758069369, −10.65149809895707071088310032074, −9.30885198775209247571672907419, −8.44125174365916490217100873391, −7.75079336187507448879479824577, −6.75576970754263677563909142211, −6.15107660397844113538707913624, −4.47473598172218021063008774984, −4.14007591655057008688066082618, −3.377826398973062360222759180006, −2.361767122528148790289838374662, −1.32601362873832507772184654533, 1.32601362873832507772184654533, 2.361767122528148790289838374662, 3.377826398973062360222759180006, 4.14007591655057008688066082618, 4.47473598172218021063008774984, 6.15107660397844113538707913624, 6.75576970754263677563909142211, 7.75079336187507448879479824577, 8.44125174365916490217100873391, 9.30885198775209247571672907419, 10.65149809895707071088310032074, 11.248394767765015672758069369, 12.21838555940824169094221333961, 12.89497858438243603385295537425, 13.6958833035159007940730151246, 14.543738810554294751857896436288, 14.97935113731995842443238277086, 15.99709480223371168987506308358, 16.190165416679817978486822692889, 17.62324229783696251913081365396, 18.77137488389543566080119264290, 19.59632647511872379771593557683, 19.93607853758708818767905584334, 20.66990356837076382369186869840, 21.54185004488475970277884980129

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