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Properties

Label 1-35e2-1225.484-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.959 + 0.280i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (−0.858 + 0.512i)2-s + (0.963 + 0.266i)3-s + (0.473 − 0.880i)4-s + (−0.963 + 0.266i)6-s + (0.0448 + 0.998i)8-s + (0.858 + 0.512i)9-s + (0.858 − 0.512i)11-s + (0.691 − 0.722i)12-s + (0.995 + 0.0896i)13-s + (−0.550 − 0.834i)16-s + (−0.473 − 0.880i)17-s − 18-s + (−0.809 − 0.587i)19-s + (−0.473 + 0.880i)22-s + (0.691 + 0.722i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (−0.858 + 0.512i)2-s + (0.963 + 0.266i)3-s + (0.473 − 0.880i)4-s + (−0.963 + 0.266i)6-s + (0.0448 + 0.998i)8-s + (0.858 + 0.512i)9-s + (0.858 − 0.512i)11-s + (0.691 − 0.722i)12-s + (0.995 + 0.0896i)13-s + (−0.550 − 0.834i)16-s + (−0.473 − 0.880i)17-s − 18-s + (−0.809 − 0.587i)19-s + (−0.473 + 0.880i)22-s + (0.691 + 0.722i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.959 + 0.280i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.645126455 + 0.2357505728i\)
\(L(\frac12)\) \(\approx\) \(1.645126455 + 0.2357505728i\)
\(L(1)\) \(\approx\) \(1.114398855 + 0.2083603279i\)
\(L(1)\) \(\approx\) \(1.114398855 + 0.2083603279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.858 + 0.512i)T \)
3 \( 1 + (0.963 + 0.266i)T \)
11 \( 1 + (0.858 - 0.512i)T \)
13 \( 1 + (0.995 + 0.0896i)T \)
17 \( 1 + (-0.473 - 0.880i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.691 + 0.722i)T \)
29 \( 1 + (0.983 - 0.178i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.691 - 0.722i)T \)
41 \( 1 + (0.936 - 0.351i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.753 - 0.657i)T \)
53 \( 1 + (-0.473 + 0.880i)T \)
59 \( 1 + (-0.550 - 0.834i)T \)
61 \( 1 + (0.134 - 0.990i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.473 - 0.880i)T \)
73 \( 1 + (0.995 - 0.0896i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.753 + 0.657i)T \)
89 \( 1 + (-0.995 + 0.0896i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−21.01307245375184811584261291067, −20.00714654220954178081062657054, −19.7311096977981930660855542410, −18.84130990395858530266080426755, −18.255875297444938634778005198852, −17.4496265482166487685208284895, −16.638496046330010196138111875177, −15.73827824308819128164040225222, −14.92691139684296112925997254326, −14.19576436208940426650828052467, −12.96070234686207805711993117873, −12.73261697076539762635149919252, −11.67240215955924018786356043039, −10.71363139622528572319426695740, −10.03787263361361913272189318737, −9.04278760020417989073930368556, −8.58331907492348410666101043133, −7.902248235101934895680918204, −6.7752009669817708311111242727, −6.3531782503711620631634160262, −4.40140473988431777553772536829, −3.75017031029687556425191970759, −2.83648529618374517132131545366, −1.81153692783181431975978193115, −1.18050603326835285698303469983, 0.92552333637415846815189464030, 1.95789691368734224626670986829, 2.951864190103154524889027268669, 4.01634750417022245988047998860, 5.010731012349962227143134979650, 6.22055021130206319522523694462, 6.89346624349936178965167415005, 7.8245419878414063240142088424, 8.62982520042772394360686052276, 9.1752341450519244975925225640, 9.76294772208962103066544366056, 11.0477761477668537500699541311, 11.25658566905362949761154945861, 12.82537443340980077470198314451, 13.75397859353705009078381414119, 14.28323001184491890010066385061, 15.1806469508852805858548328144, 15.78166024270287118405472937545, 16.45893810239378429039696931701, 17.30331325744218769239530055090, 18.24690748512092300055695105439, 18.849820086946218342521499916822, 19.7159574270109105970858284102, 20.0193976159857252061556714877, 21.079608124023956963585957945783

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