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Properties

Label 1-1003-1003.1002-r1-0-0
Degree $1$
Conductor $1003$
Sign $1$
Analytic cond. $107.787$
Root an. cond. $107.787$
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 − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s − 29-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 − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $1$
Analytic conductor: \(107.787\)
Root analytic conductor: \(107.787\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1003} (1002, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1003,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4399532706\)
\(L(\frac12)\) \(\approx\) \(0.4399532706\)
\(L(1)\) \(\approx\) \(0.3967887931\)
\(L(1)\) \(\approx\) \(0.3967887931\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
59 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
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 \( 1 + T \)
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

−21.546525129217370640872665519165, −20.325254648321915671631696015761, −19.67875317504041085757091858181, −19.056641201829355090118740248377, −18.43230033042976976898629012166, −17.39912367975082974146792297851, −16.66071992683849151863505246581, −16.36044275161020047156477203239, −15.36976323922736262711340642415, −14.828719056229163157256426823281, −13.18058412159619486084046349568, −12.255709975481852946317732715909, −11.73916106302579209453775238336, −11.157133485580748490288402272006, −9.99989160279080179270181666490, −9.58355614598685994452339330847, −8.52207860752553281873803978308, −7.26107344656632780112159510019, −7.021959618701832162318027238465, −6.08385836649451887306201638365, −4.97150927930841979182780340619, −3.786372553531169199186648989273, −2.87576197328183559076307936749, −1.31600486414300555441241875960, −0.418736286285996742983971842692, 0.418736286285996742983971842692, 1.31600486414300555441241875960, 2.87576197328183559076307936749, 3.786372553531169199186648989273, 4.97150927930841979182780340619, 6.08385836649451887306201638365, 7.021959618701832162318027238465, 7.26107344656632780112159510019, 8.52207860752553281873803978308, 9.58355614598685994452339330847, 9.99989160279080179270181666490, 11.157133485580748490288402272006, 11.73916106302579209453775238336, 12.255709975481852946317732715909, 13.18058412159619486084046349568, 14.828719056229163157256426823281, 15.36976323922736262711340642415, 16.36044275161020047156477203239, 16.66071992683849151863505246581, 17.39912367975082974146792297851, 18.43230033042976976898629012166, 19.056641201829355090118740248377, 19.67875317504041085757091858181, 20.325254648321915671631696015761, 21.546525129217370640872665519165

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