↑

Properties

Label 1-1205-1205.1204-r0-0-0
Degree $1$
Conductor $1205$
Sign $1$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 11-s − 12-s + 13-s − 14-s + 16-s + 17-s − 18-s − 19-s − 21-s + 22-s + 23-s + 24-s − 26-s − 27-s + 28-s + 29-s − 31-s − 32-s + 33-s − 34-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 11-s − 12-s + 13-s − 14-s + 16-s + 17-s − 18-s − 19-s − 21-s + 22-s + 23-s + 24-s − 26-s − 27-s + 28-s + 29-s − 31-s − 32-s + 33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $1$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1205} (1204, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8095007876\)
\(L(\frac12)\) \(\approx\) \(0.8095007876\)
\(L(1)\) \(\approx\) \(0.6329648390\)
\(L(1)\) \(\approx\) \(0.6329648390\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
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

−21.08925672442188265248654302869, −20.65365940588051427262331166294, −19.30231452248477439501410642921, −18.674643856553540677355174725065, −17.96856963895190266853892352315, −17.56359606969522537274875483654, −16.604114225646914208076530691428, −16.10702600051602449334524939875, −15.2441903852685902014115141859, −14.47742253186643201318294496478, −13.08128726431376142156629522721, −12.43070478605544922068164978030, −11.39781100548197072007255083515, −10.916319526167756722852654574958, −10.40355314437758346281336718929, −9.37887968308001554785189224840, −8.31977922532332434889219991982, −7.75656612454626141847618192860, −6.84123334487362335542527579247, −5.89654103618655572324687162233, −5.24630022104214798578499034610, −4.1386748919721530486195957873, −2.76836933691101336275667381838, −1.61777977016581667613663374130, −0.81286572425811051888019446950, 0.81286572425811051888019446950, 1.61777977016581667613663374130, 2.76836933691101336275667381838, 4.1386748919721530486195957873, 5.24630022104214798578499034610, 5.89654103618655572324687162233, 6.84123334487362335542527579247, 7.75656612454626141847618192860, 8.31977922532332434889219991982, 9.37887968308001554785189224840, 10.40355314437758346281336718929, 10.916319526167756722852654574958, 11.39781100548197072007255083515, 12.43070478605544922068164978030, 13.08128726431376142156629522721, 14.47742253186643201318294496478, 15.2441903852685902014115141859, 16.10702600051602449334524939875, 16.604114225646914208076530691428, 17.56359606969522537274875483654, 17.96856963895190266853892352315, 18.674643856553540677355174725065, 19.30231452248477439501410642921, 20.65365940588051427262331166294, 21.08925672442188265248654302869

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