↑

Properties

Label 1-1588-1588.1587-r1-0-0
Degree $1$
Conductor $1588$
Sign $1$
Analytic cond. $170.654$
Root an. cond. $170.654$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 11-s − 13-s + 15-s − 17-s − 19-s − 21-s − 23-s + 25-s − 27-s + 29-s − 31-s + 33-s − 35-s + 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s + 51-s − 53-s + 55-s + ⋯
L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 11-s − 13-s + 15-s − 17-s − 19-s − 21-s − 23-s + 25-s − 27-s + 29-s − 31-s + 33-s − 35-s + 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s + 51-s − 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1588 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1588\)    =    \(2^{2} \cdot 397\)
Sign: $1$
Analytic conductor: \(170.654\)
Root analytic conductor: \(170.654\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1588} (1587, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1588,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1778780114\)
\(L(\frac12)\) \(\approx\) \(0.1778780114\)
\(L(1)\) \(\approx\) \(0.4730160466\)
\(L(1)\) \(\approx\) \(0.4730160466\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
397 \( 1 \)
good3 \( 1 - T \)
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 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.192424103230421595508559866226, −19.64463951802134287941914802802, −18.54417961708297920908434835014, −18.1301134453278201989491501804, −17.33542423861547995590952244735, −16.61447188373789548214132567100, −15.811393652802704819059631099472, −15.19938953751277292644211697632, −14.53521705139856549693821331152, −13.30837540598780243976817154067, −12.55720157913539442914658337252, −11.88901446293079093000085395641, −11.198648545099174251678013747592, −10.6528753643813863098705116581, −9.84362414517346067908679813711, −8.5007757245359223960785732199, −7.91646920723793898796795416835, −7.15141807008759764878138314630, −6.329133215203240866533766778797, −5.13610810880029162385955188409, −4.707545493101938764569098013489, −3.98627405282904727632483715314, −2.56590074296139885123800207005, −1.609669498838646197208208809433, −0.18579358431866176065191170685, 0.18579358431866176065191170685, 1.609669498838646197208208809433, 2.56590074296139885123800207005, 3.98627405282904727632483715314, 4.707545493101938764569098013489, 5.13610810880029162385955188409, 6.329133215203240866533766778797, 7.15141807008759764878138314630, 7.91646920723793898796795416835, 8.5007757245359223960785732199, 9.84362414517346067908679813711, 10.6528753643813863098705116581, 11.198648545099174251678013747592, 11.88901446293079093000085395641, 12.55720157913539442914658337252, 13.30837540598780243976817154067, 14.53521705139856549693821331152, 15.19938953751277292644211697632, 15.811393652802704819059631099472, 16.61447188373789548214132567100, 17.33542423861547995590952244735, 18.1301134453278201989491501804, 18.54417961708297920908434835014, 19.64463951802134287941914802802, 20.192424103230421595508559866226

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