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Properties

Label 1-275-275.212-r1-0-0
Degree $1$
Conductor $275$
Sign $0.163 - 0.986i$
Analytic cond. $29.5528$
Root an. cond. $29.5528$
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  + i·2-s + (−0.951 − 0.309i)3-s − 4-s + (0.309 − 0.951i)6-s + (0.587 − 0.809i)7-s − i·8-s + (0.809 + 0.587i)9-s + (0.951 + 0.309i)12-s + (−0.587 + 0.809i)13-s + (0.809 + 0.587i)14-s + 16-s + (0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s − 19-s + (−0.809 + 0.587i)21-s + ⋯
L(s)  = 1  + i·2-s + (−0.951 − 0.309i)3-s − 4-s + (0.309 − 0.951i)6-s + (0.587 − 0.809i)7-s − i·8-s + (0.809 + 0.587i)9-s + (0.951 + 0.309i)12-s + (−0.587 + 0.809i)13-s + (0.809 + 0.587i)14-s + 16-s + (0.951 + 0.309i)17-s + (−0.587 + 0.809i)18-s − 19-s + (−0.809 + 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ 0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3889808469 - 0.3296588955i\)
\(L(\frac12)\) \(\approx\) \(0.3889808469 - 0.3296588955i\)
\(L(1)\) \(\approx\) \(0.6473681740 + 0.1601266359i\)
\(L(1)\) \(\approx\) \(0.6473681740 + 0.1601266359i\)

Euler product

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

−25.80986754816151274782900711804, −24.55310868345376570960842526766, −23.547177307905492012745055566235, −22.706014953648452402812232925480, −21.95097161924774875792316055179, −21.19059412767653839028870022203, −20.48794344033620560564741565594, −19.075316320785406283825971764563, −18.4421460509265465812397961670, −17.47058638077083309103956279160, −16.82318996503114078220562314025, −15.292616721227409606012467574804, −14.61622912184518964157403206846, −13.04532085477655568188910990470, −12.33632814029077192845263868951, −11.5074393161647187701104299710, −10.70043766112244228489823846391, −9.76924659057822530700835036001, −8.768383580890492859178532891647, −7.461780278127672594047072655918, −5.67343141895357188892410050282, −5.16198795931315623817045984445, −3.95051126569848449037439138873, −2.579600374750635410475577517500, −1.19741418470949548557268128487, 0.20672277207647512289184518760, 1.56429285332506418436291845551, 3.96318200192966649047675822567, 4.85504339229042894080605486229, 5.812084492368828857905056384942, 7.00645264693076368247553244326, 7.48945975842320701890817660994, 8.80271680749503714270208154578, 10.11516466065972040240370293134, 10.99802642565557735733094834030, 12.28310421313761709741928739012, 13.1417927810556844701487133760, 14.2352905242322682755112174866, 15.00580520428238998022055365446, 16.43096344202658304607832224259, 16.87174644560245869031219061387, 17.570097876623525766927410860652, 18.617581716569712958274979200450, 19.3666083220955633501672102517, 21.050499241480112756666723384856, 21.8390311016234619825151631904, 22.93637533311525556694260716949, 23.561411193535184518393764466896, 24.1480543158381584293958400878, 25.06069208182010281342577684162

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