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Properties

Label 1-2869-2869.2868-r0-0-0
Degree $1$
Conductor $2869$
Sign $1$
Analytic cond. $13.3235$
Root an. cond. $13.3235$
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 + 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 + 17-s − 18-s + 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s + 27-s − 28-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2869 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2869 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2869\)    =    \(19 \cdot 151\)
Sign: $1$
Analytic conductor: \(13.3235\)
Root analytic conductor: \(13.3235\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2869} (2868, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2869,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.144167851\)
\(L(\frac12)\) \(\approx\) \(2.144167851\)
\(L(1)\) \(\approx\) \(1.253837637\)
\(L(1)\) \(\approx\) \(1.253837637\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
151 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
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

−19.04729917651657458215061708107, −18.58239791357776173946725337976, −17.94799121903382758732579521454, −16.99246674531451223251381748438, −16.446025530071721057176966154373, −15.822464346123037764920342193686, −14.98903792222003427842965161183, −14.18214402202253183246232623363, −13.70510976148584511171089572258, −12.68267990035331190631536501650, −12.24651317785579421340223534710, −10.97830081482891228560842527574, −10.30374341120787395420526978750, −9.587273439344632181466131218521, −9.18939913722543067989452049192, −8.651393517444338340213367795863, −7.62110753179500999627015320817, −6.97431797236074764554505690387, −6.136211210553696291043988240369, −5.67350425082180334615024821389, −3.90741794994611664731226927693, −3.429655158002939400894175931536, −2.48523695951545252529316216999, −1.71941811864051970068761280480, −0.99730785003915228742772139618, 0.99730785003915228742772139618, 1.71941811864051970068761280480, 2.48523695951545252529316216999, 3.429655158002939400894175931536, 3.90741794994611664731226927693, 5.67350425082180334615024821389, 6.136211210553696291043988240369, 6.97431797236074764554505690387, 7.62110753179500999627015320817, 8.651393517444338340213367795863, 9.18939913722543067989452049192, 9.587273439344632181466131218521, 10.30374341120787395420526978750, 10.97830081482891228560842527574, 12.24651317785579421340223534710, 12.68267990035331190631536501650, 13.70510976148584511171089572258, 14.18214402202253183246232623363, 14.98903792222003427842965161183, 15.822464346123037764920342193686, 16.446025530071721057176966154373, 16.99246674531451223251381748438, 17.94799121903382758732579521454, 18.58239791357776173946725337976, 19.04729917651657458215061708107

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