Properties

Label 1-1033-1033.1032-r0-0-0
Degree $1$
Conductor $1033$
Sign $1$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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 − 13-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 − 26-s + 27-s + 28-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 − 13-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 − 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $1$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1033} (1032, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.963929616\)
\(L(\frac12)\) \(\approx\) \(3.963929616\)
\(L(1)\) \(\approx\) \(2.517852251\)
\(L(1)\) \(\approx\) \(2.517852251\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
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

−21.45808382779393398213227846294, −20.67457407969200477224594835670, −20.33174312133962866920159905894, −19.43244490890856913209032534802, −18.77154690479651963394004286002, −17.76396403603661915339303261028, −16.40246563763102931932098166676, −15.85755531953847012337881879946, −15.00352710771884096250618920707, −14.54419794513183726709919201147, −13.87428688274220257027142141237, −12.868772019265497311226088206841, −12.167368479151846916339151943681, −11.45407698832281336874744781153, −10.487172839864718332618844564739, −9.61033125669792573566200941329, −8.11895886409747040562712658966, −7.728279079810200086134797658569, −7.28552194238342835712097160231, −5.71768142066804416362431698172, −4.77062758128312056759948333089, −4.19469010108334357398078023047, −3.13297778280283103907163159543, −2.496190219884075234644007637956, −1.32333741042229183244633326411, 1.32333741042229183244633326411, 2.496190219884075234644007637956, 3.13297778280283103907163159543, 4.19469010108334357398078023047, 4.77062758128312056759948333089, 5.71768142066804416362431698172, 7.28552194238342835712097160231, 7.728279079810200086134797658569, 8.11895886409747040562712658966, 9.61033125669792573566200941329, 10.487172839864718332618844564739, 11.45407698832281336874744781153, 12.167368479151846916339151943681, 12.868772019265497311226088206841, 13.87428688274220257027142141237, 14.54419794513183726709919201147, 15.00352710771884096250618920707, 15.85755531953847012337881879946, 16.40246563763102931932098166676, 17.76396403603661915339303261028, 18.77154690479651963394004286002, 19.43244490890856913209032534802, 20.33174312133962866920159905894, 20.67457407969200477224594835670, 21.45808382779393398213227846294

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