Properties

Label 1-129-129.128-r0-0-0
Degree $1$
Conductor $129$
Sign $1$
Analytic cond. $0.599073$
Root an. cond. $0.599073$
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 + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s − 19-s + 20-s − 22-s − 23-s + 25-s + 26-s − 28-s + 29-s + 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s − 19-s + 20-s − 22-s − 23-s + 25-s + 26-s − 28-s + 29-s + 31-s + 32-s − 34-s − 35-s − 37-s − 38-s + 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $1$
Analytic conductor: \(0.599073\)
Root analytic conductor: \(0.599073\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{129} (128, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 129,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.981020375\)
\(L(\frac12)\) \(\approx\) \(1.981020375\)
\(L(1)\) \(\approx\) \(1.835836955\)
\(L(1)\) \(\approx\) \(1.835836955\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 \)
good2 \( 1 + T \)
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 \( 1 - T \)
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

−28.92503322763303281849392160292, −28.30346814709051935989006213871, −26.175562985130485956339608171218, −25.76011720626284665983809601606, −24.72517076905428831589850174120, −23.58333578895241070728058294509, −22.710600015160979431809668071198, −21.717030616432443637056296311, −20.96796244708498986455341065005, −19.919152580134126358970449134840, −18.65326493193326148316260679010, −17.372559097162408186160883128543, −16.10446700917013154437614951768, −15.423827187306917393909829844, −13.86138053952253514514626317218, −13.31420313500417711586281266252, −12.41750639880045620278095023399, −10.82209265347298076555966000015, −10.03163214254246147181169030819, −8.42634781748281209173830364905, −6.62255494014020266337495914010, −6.058358901333976442032144499644, −4.69999004651383639972471567304, −3.192000650645312588512587572055, −2.03656834886337276049517659329, 2.03656834886337276049517659329, 3.192000650645312588512587572055, 4.69999004651383639972471567304, 6.058358901333976442032144499644, 6.62255494014020266337495914010, 8.42634781748281209173830364905, 10.03163214254246147181169030819, 10.82209265347298076555966000015, 12.41750639880045620278095023399, 13.31420313500417711586281266252, 13.86138053952253514514626317218, 15.423827187306917393909829844, 16.10446700917013154437614951768, 17.372559097162408186160883128543, 18.65326493193326148316260679010, 19.919152580134126358970449134840, 20.96796244708498986455341065005, 21.717030616432443637056296311, 22.710600015160979431809668071198, 23.58333578895241070728058294509, 24.72517076905428831589850174120, 25.76011720626284665983809601606, 26.175562985130485956339608171218, 28.30346814709051935989006213871, 28.92503322763303281849392160292

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