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Properties

Label 1-5e2-25.11-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

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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: $\chi_{25} (11, \cdot )$
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 \( 1 + (-0.809 - 0.587i)T \)
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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

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

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