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Properties

Label 1-5e2-25.8-r1-0-0
Degree $1$
Conductor $25$
Sign $0.248 + 0.968i$
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  + (0.587 + 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (0.587 − 0.809i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + i·18-s + (−0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.951 + 0.309i)3-s + (−0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (0.587 − 0.809i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + i·18-s + (−0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.248 + 0.968i$
Analytic conductor: \(2.68662\)
Root analytic conductor: \(2.68662\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 25,\ (1:\ ),\ 0.248 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.698892521 + 1.317796120i\)
\(L(\frac12)\) \(\approx\) \(1.698892521 + 1.317796120i\)
\(L(1)\) \(\approx\) \(1.502935290 + 0.8146263744i\)
\(L(1)\) \(\approx\) \(1.502935290 + 0.8146263744i\)

Euler product

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

−37.83914813124235530266748124924, −36.95296793427771887093297939565, −35.66072480695777225366142233666, −33.78789509266728952939908101014, −32.02657057510138798061776627674, −31.523862125357787500857504281564, −30.32285426605063366456286951633, −29.076146298482492391157443057720, −27.72683394011576383827045427617, −26.05638480343179985683091206165, −24.604934083227172991538570894799, −23.4153931595628825998142240338, −21.53611502144450401795286496697, −20.829278426493166479653774278, −19.10775540941618199150071974083, −18.53767676286194792550544945503, −15.64843473466794840508725022084, −14.34576141050023920624390795965, −13.1257342764403463460291410638, −11.787506531645565541452875646307, −9.83699969076060357130631425927, −8.35729297605591647017677515242, −5.85314952260678614135089002329, −3.60948880523282830801389596731, −2.00962811207997792106880625555, 3.233418702909588958935034996908, 4.82460368351358800972059216098, 7.162107251158793135539960620329, 8.36320321565733722042263037584, 10.281685234816200883568263210570, 12.86075528096509909418139352687, 13.927268770752639087339477676644, 15.203437904668665289310507087237, 16.396276979937542350697677652775, 18.092878207712702470918655031591, 20.15907046005617239439438131795, 21.14235793288864123145726798648, 22.80133874512205780707630217717, 24.01802416462035286251035286780, 25.53432201050028880350409263436, 26.23558771727709495506183551581, 27.560065896921924650537764698394, 29.98311919115548044573858749999, 30.8805355043744549490832883920, 32.24413557918853681821285672084, 32.97860683977653684130801577331, 34.27586574913220758342749273218, 35.975598991709876343890990784535, 36.74735205333305172005801626122, 38.59889367890269848499165209757

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