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Properties

Label 1-35e2-1225.939-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: $\chi_{1225} (939, \cdot )$
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

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

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