↑

Properties

Label 1-5e2-25.23-r1-0-0
Degree $1$
Conductor $25$
Sign $-0.187 - 0.982i$
Analytic cond. $2.68662$
Root an. cond. $2.68662$
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  + (−0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s − i·7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)12-s + (−0.951 + 0.309i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + i·18-s + (0.809 − 0.587i)19-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s − i·7-s + (−0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + (0.309 − 0.951i)11-s + (0.951 − 0.309i)12-s + (−0.951 + 0.309i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + i·18-s + (0.809 − 0.587i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(2.68662\)
Root analytic conductor: \(2.68662\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (1:\ ),\ -0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6447300201 - 0.7793447164i\)
\(L(\frac12)\) \(\approx\) \(0.6447300201 - 0.7793447164i\)
\(L(1)\) \(\approx\) \(0.7422581621 - 0.4449586825i\)
\(L(1)\) \(\approx\) \(0.7422581621 - 0.4449586825i\)

Euler product

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

−38.18032658172820478651300422064, −37.23956141102694371665256472252, −36.07410042646804915281895378680, −34.60708824148501390198007573854, −33.53029009449851364147581123610, −32.290098339412004849073389491877, −30.93647437606106020903457044129, −29.020891228401326118514567875090, −27.76592015262657287458855724947, −26.937370590951947319496327143508, −25.44479545347519047315252707127, −24.81110471234110254220674287560, −22.554738471618482718543933796679, −20.94263438758535628752522516288, −19.79751353288739774116964077882, −18.43659889685501390907522205087, −16.83308497875659815585739629775, −15.465058561758441375706318282157, −14.57262733679717144719959217015, −11.918653617674503527194083085845, −10.04127195440645353175684419329, −9.10948746674385242248092493874, −7.546235426316997947360196195419, −5.271666807742885535512824955853, −2.53697086841699069617471037383, 1.0898559021293950958244825664, 3.20843322220895894998374361343, 6.7871873749236234618213614700, 7.97748115239263831924215192555, 9.49445241590965497549378522766, 11.25230354375311149672311815375, 12.84634102074031597494371085783, 14.43561056518546282494300754755, 16.5542441460623700876898786912, 17.75573810934767604369819677239, 19.25651196255319853696278310857, 19.93329096523997007659774814694, 21.466182241672750826460666530773, 23.74223102612164696738014858691, 24.87292692043499896421040020616, 26.2484739627926259015310592243, 27.09533535067808450296134328958, 29.02361931835678477196160147047, 29.800000703946848453403359455246, 30.918100632855257343805854974210, 32.64296172577014671770666622190, 34.45337264094638941686459261460, 35.42973467380984508115249661878, 36.71625410938427560900799928228, 37.16291300369222072128001830107

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