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Properties

Label 1-525-525.83-r1-0-0
Degree $1$
Conductor $525$
Sign $0.998 + 0.0627i$
Analytic cond. $56.4190$
Root an. cond. $56.4190$
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 + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)11-s + (−0.587 + 0.809i)13-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.951 − 0.309i)22-s + (0.587 + 0.809i)23-s + 26-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + i·32-s + (−0.809 − 0.587i)34-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)8-s + (0.809 − 0.587i)11-s + (−0.587 + 0.809i)13-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.951 − 0.309i)22-s + (0.587 + 0.809i)23-s + 26-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + i·32-s + (−0.809 − 0.587i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.998 + 0.0627i$
Analytic conductor: \(56.4190\)
Root analytic conductor: \(56.4190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (1:\ ),\ 0.998 + 0.0627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.369527621 + 0.04303913938i\)
\(L(\frac12)\) \(\approx\) \(1.369527621 + 0.04303913938i\)
\(L(1)\) \(\approx\) \(0.8290776493 - 0.1809572991i\)
\(L(1)\) \(\approx\) \(0.8290776493 - 0.1809572991i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
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

−23.41656405004259096994813911334, −22.612773147135104788834745694402, −21.86311815396312647205706392639, −20.48251905157698462256562877650, −19.762426188182103801493348969704, −19.03838092652530008421797399115, −17.96107510739607496234426493514, −17.39893105162942869874906729545, −16.58267741605112369204233389502, −15.69112221255941593052466761514, −14.738449523007822309456658056088, −14.31306957582335121786099083220, −13.02182022848790959514954954730, −12.12215013066569141965215412611, −10.84075460464494212051498479483, −10.05996352386401956495826321263, −9.17856275404338798253076915532, −8.33270908824990700674821168055, −7.2584679424777563342413072107, −6.66848633239908112583566057209, −5.40566663937550158222404265473, −4.708214331861447105663081999958, −3.23247195658494246677763044695, −1.70777788294715214994521640723, −0.55345269842328916889091163875, 0.91520515748236910264715864295, 1.91193977729216927316279102123, 3.20319664313563037287439148879, 3.9884699896689046897027542397, 5.23200659587024658038201457708, 6.5907096377626591010633276077, 7.627071890612506960157036017259, 8.49222679055530505865202166760, 9.571918363667423222620297861909, 9.996699919955747792690846066807, 11.48193247746920323785862268544, 11.686577483100358261590898036730, 12.78029612712632297958718055958, 13.81983394026466436394622302654, 14.528726332361787423166386281832, 15.93992203148756014698317170532, 16.8622688331737843681314212663, 17.255544826226601416578435044403, 18.60691258417686448288280348967, 19.017424652106209713611408353767, 19.82364723373339847360703477343, 20.84011435800796547960957028069, 21.421850542468059210517912566078, 22.30613290213756016990468968795, 23.048846740134971791947031033481

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