↑

Properties

Label 1-5e2-25.16-r0-0-0
Degree $1$
Conductor $25$
Sign $-0.187 + 0.982i$
Analytic cond. $0.116099$
Root an. cond. $0.116099$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + 7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + 18-s + (−0.809 − 0.587i)19-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + 7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + 18-s + (−0.809 − 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(0.116099\)
Root analytic conductor: \(0.116099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (0:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4161715553 + 0.5030649925i\)
\(L(\frac12)\) \(\approx\) \(0.4161715553 + 0.5030649925i\)
\(L(1)\) \(\approx\) \(0.6893399648 + 0.5231126992i\)
\(L(1)\) \(\approx\) \(0.6893399648 + 0.5231126992i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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

−38.12939810570295828152552956012, −36.880668857792598348449904327355, −35.73198325910141512095877508369, −34.24069041082059349109863905510, −32.94674784639787323852551624921, −31.21166078246638998204563090651, −30.26378473632545666009257209682, −29.15124546703276676786113430223, −28.072187508295193438693183365737, −26.90110304856066464864837213249, −24.40561667021324992168570779265, −23.59411718769541543306838299218, −22.12041304351818239751594381095, −21.06391048239070500440409047022, −19.269158710552615347615334904364, −18.24498443316952626254182308585, −16.85894097142759972300817021446, −14.458708726440136328635402357078, −13.09628717870442735880585683571, −11.60858925336675440225000122594, −10.80940300610891623952709287997, −8.53698760265408212607732567885, −6.13806006100624507749648070034, −4.44236786101502659110732747155, −1.73898283150814053848261366782, 4.28610390947048074437857317947, 5.51856744164026103521151399457, 7.27547535065183704378652478450, 9.1831895622885488865799593965, 11.126100715771593196211246841374, 12.79233874209598185729631581975, 14.736037675373766708396497009809, 15.68450245880877727361580951041, 17.31766117422789019033416706441, 17.94121574901487487502193832433, 20.65923799347786472530655770864, 22.01517994744368483296669511753, 23.08907072587407899899754917730, 24.24129613212810482160429898772, 25.707150603530566212395836461275, 27.26615192230849273250670174772, 27.9311103895794250828035657699, 29.98068278860309287831214022722, 31.357186425066558271287382576383, 32.86993985379066843433646526869, 33.5977783290343765356127712823, 34.65102533550792200966772282565, 35.81110296284364746257822303034, 37.47967765264783867439693239396, 39.11250341028089759025193620043

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