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Properties

Label 1-5e2-25.12-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

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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: $\chi_{25} (12, \cdot )$
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 \( 1 + (-0.587 - 0.809i)T \)
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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

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

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