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Properties

Label 1-35e2-1225.309-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.430 - 0.902i$
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  + (−0.134 + 0.990i)2-s + (−0.753 − 0.657i)3-s + (−0.963 − 0.266i)4-s + (0.753 − 0.657i)6-s + (0.393 − 0.919i)8-s + (0.134 + 0.990i)9-s + (0.134 − 0.990i)11-s + (0.550 + 0.834i)12-s + (0.691 − 0.722i)13-s + (0.858 + 0.512i)16-s + (0.963 − 0.266i)17-s − 18-s + (−0.809 − 0.587i)19-s + (0.963 + 0.266i)22-s + (0.550 − 0.834i)23-s + (−0.900 + 0.433i)24-s + ⋯
L(s)  = 1  + (−0.134 + 0.990i)2-s + (−0.753 − 0.657i)3-s + (−0.963 − 0.266i)4-s + (0.753 − 0.657i)6-s + (0.393 − 0.919i)8-s + (0.134 + 0.990i)9-s + (0.134 − 0.990i)11-s + (0.550 + 0.834i)12-s + (0.691 − 0.722i)13-s + (0.858 + 0.512i)16-s + (0.963 − 0.266i)17-s − 18-s + (−0.809 − 0.587i)19-s + (0.963 + 0.266i)22-s + (0.550 − 0.834i)23-s + (−0.900 + 0.433i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6814708202 - 0.4300269727i\)
\(L(\frac12)\) \(\approx\) \(0.6814708202 - 0.4300269727i\)
\(L(1)\) \(\approx\) \(0.7264528543 + 0.01894287063i\)
\(L(1)\) \(\approx\) \(0.7264528543 + 0.01894287063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.134 + 0.990i)T \)
3 \( 1 + (-0.753 - 0.657i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
13 \( 1 + (0.691 - 0.722i)T \)
17 \( 1 + (0.963 - 0.266i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.550 - 0.834i)T \)
29 \( 1 + (-0.0448 + 0.998i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.550 + 0.834i)T \)
41 \( 1 + (-0.995 - 0.0896i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.983 + 0.178i)T \)
53 \( 1 + (0.963 + 0.266i)T \)
59 \( 1 + (0.858 + 0.512i)T \)
61 \( 1 + (0.936 + 0.351i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (-0.963 - 0.266i)T \)
73 \( 1 + (0.691 + 0.722i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.983 - 0.178i)T \)
89 \( 1 + (-0.691 - 0.722i)T \)
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

−21.235373491254228752716147666671, −20.76553078803086495151407166997, −19.81584860447309950112405350766, −18.98432682919988224496493140619, −18.22841955830880627317977641322, −17.485623559905017724157476993484, −16.84142277650510140823169912021, −16.10638478115048320362399326073, −14.93670849543169051577638319324, −14.42099196545410358746655184538, −13.18279612932841581155113801461, −12.52869016059761299356557997895, −11.720269288049139411204510559813, −11.17832486883774524051957869403, −10.25712017635297663916598660979, −9.72081894410096019707943492436, −8.97634357422927596634468243119, −7.953099122055452869299098662136, −6.786805544655022519822884313444, −5.73603506660844677668586051635, −4.91237389936666393553256094183, −4.01108935858827270646923089388, −3.49969705866120927215357946373, −2.05042150326510236827374597197, −1.16715286026852640204945959502, 0.46358057233166255275816151619, 1.3329915134240939306309473626, 2.95333386438666005408777443894, 4.11120020947866139626263422041, 5.28581269513550625427244071611, 5.70422382080919886638912285743, 6.62075584681960553756182209127, 7.22632142032498880907613299048, 8.31796346397446771945349118999, 8.668589410394648374923558026040, 10.0395878409432768930482474947, 10.7708334530563563418405906339, 11.59319182597241695046654692521, 12.794492799151620864232606578092, 13.15759725282593194072016883966, 14.05179515238369053493456980506, 14.86000802403213716008044930215, 15.80784234764857820246701496311, 16.6277859262509513340669273774, 16.9232164444590308957812879309, 17.9248515201555120355479397028, 18.593834018086064282415047203181, 18.98195793074176444297250724143, 20.02626809209683063177509098661, 21.23648851089659692625107429820

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