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Properties

Label 1-275-275.13-r0-0-0
Degree $1$
Conductor $275$
Sign $-0.462 + 0.886i$
Analytic cond. $1.27709$
Root an. cond. $1.27709$
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 + (−0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s − 6-s + (−0.951 + 0.309i)7-s + (0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (−0.951 − 0.309i)12-s + (0.587 + 0.809i)13-s − 14-s + (0.309 + 0.951i)16-s + i·17-s + (0.951 − 0.309i)18-s + (−0.809 + 0.587i)19-s + (0.809 − 0.587i)21-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s − 6-s + (−0.951 + 0.309i)7-s + (0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (−0.951 − 0.309i)12-s + (0.587 + 0.809i)13-s − 14-s + (0.309 + 0.951i)16-s + i·17-s + (0.951 − 0.309i)18-s + (−0.809 + 0.587i)19-s + (0.809 − 0.587i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.462 + 0.886i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.462 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6937141655 + 1.144112884i\)
\(L(\frac12)\) \(\approx\) \(0.6937141655 + 1.144112884i\)
\(L(1)\) \(\approx\) \(1.068456115 + 0.6197807266i\)
\(L(1)\) \(\approx\) \(1.068456115 + 0.6197807266i\)

Euler product

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

−25.15289334887991196986640065841, −24.152458452954366555076925763445, −23.4198611021255917413510754958, −22.5017095926510915437959369290, −22.26153364696088997386425262101, −20.95018513760148641212726705204, −20.04528490773338512371077242466, −19.06340952845137467653697701216, −18.16202016797627652495121869896, −16.87301045943503455050210551872, −16.0595748690170822391758960858, −15.31944648688230944992440298823, −13.84032644669122709014255473948, −13.07418353480056960399392002605, −12.422490344701050632333166720851, −11.330855710494902573388488831420, −10.56833770296733181694398746113, −9.60771720771436529211442995357, −7.66296387117144478662631571986, −6.59360312260117409276768989227, −5.93736280704839690787913616234, −4.82075738051028375342996005384, −3.69874557328928529560446133594, −2.387850608258195013207933636808, −0.73998390110998252460427002139, 1.930587779271534818844469995774, 3.64102709753921795390975642443, 4.294060104985376505111211140087, 5.84831857605795271005322006807, 6.1389820972686939084565566074, 7.27099024204711486582175361363, 8.78565277925966198223045887351, 10.15557054533256285348189345763, 11.06974615391026464082950449805, 12.152027756813973413945775391962, 12.71240953352495473583878947574, 13.79776895280521991538564081647, 15.03218570981332458631431467310, 15.857813716928335845459002554092, 16.532806213226891324034371879397, 17.334568530502523002816310466060, 18.638823163478287290214227355097, 19.678682553370256188768050502992, 21.03057281069171742373859452398, 21.63977294440585711989483905302, 22.40486663434429862539383762534, 23.29325272309224178487573534370, 23.78068760687589800985417393305, 24.93042571669150312199089681848, 25.91576492222919931404427193554

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