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Properties

Label 1-35e2-1225.254-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.873 + 0.486i$
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  + (−0.525 + 0.850i)2-s + (−0.0149 − 0.999i)3-s + (−0.447 − 0.894i)4-s + (0.858 + 0.512i)6-s + (0.995 + 0.0896i)8-s + (−0.999 + 0.0299i)9-s + (−0.999 − 0.0299i)11-s + (−0.887 + 0.460i)12-s + (−0.983 + 0.178i)13-s + (−0.599 + 0.800i)16-s + (−0.998 + 0.0598i)17-s + (0.5 − 0.866i)18-s + (−0.978 + 0.207i)19-s + (0.550 − 0.834i)22-s + (0.842 − 0.538i)23-s + (0.0747 − 0.997i)24-s + ⋯
L(s)  = 1  + (−0.525 + 0.850i)2-s + (−0.0149 − 0.999i)3-s + (−0.447 − 0.894i)4-s + (0.858 + 0.512i)6-s + (0.995 + 0.0896i)8-s + (−0.999 + 0.0299i)9-s + (−0.999 − 0.0299i)11-s + (−0.887 + 0.460i)12-s + (−0.983 + 0.178i)13-s + (−0.599 + 0.800i)16-s + (−0.998 + 0.0598i)17-s + (0.5 − 0.866i)18-s + (−0.978 + 0.207i)19-s + (0.550 − 0.834i)22-s + (0.842 − 0.538i)23-s + (0.0747 − 0.997i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.873 + 0.486i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.873 + 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6451369780 + 0.1674077550i\)
\(L(\frac12)\) \(\approx\) \(0.6451369780 + 0.1674077550i\)
\(L(1)\) \(\approx\) \(0.6380450836 + 0.04306455050i\)
\(L(1)\) \(\approx\) \(0.6380450836 + 0.04306455050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.525 + 0.850i)T \)
3 \( 1 + (-0.0149 - 0.999i)T \)
11 \( 1 + (-0.999 - 0.0299i)T \)
13 \( 1 + (-0.983 + 0.178i)T \)
17 \( 1 + (-0.998 + 0.0598i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.842 - 0.538i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.887 + 0.460i)T \)
41 \( 1 + (0.753 + 0.657i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.925 - 0.379i)T \)
53 \( 1 + (0.447 + 0.894i)T \)
59 \( 1 + (0.992 + 0.119i)T \)
61 \( 1 + (0.712 + 0.701i)T \)
67 \( 1 + (-0.669 - 0.743i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (0.646 - 0.762i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.134 - 0.990i)T \)
89 \( 1 + (-0.337 - 0.941i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−21.0800356282612201461936993068, −20.4389543669879872355753417017, −19.45487578317728103703878720008, −19.162454883335354620747430400582, −17.640827939546065469150905595428, −17.58281891318284141095293668172, −16.57701206481160324800260493958, −15.710923437896316792528067321807, −15.10332428638290879585125383320, −14.015625778532813738160287541131, −13.14911149797856531435488131221, −12.390663561146245204150161205247, −11.39178128765929967201758506862, −10.78843936045862533910349016429, −10.12716939356430729852200884743, −9.4112250397752932984372446162, −8.62144281017083842385930156687, −7.87930586531015273541531799151, −6.806602257223871343506122405516, −5.39214356407871100899822753112, −4.64944640433969008466124080830, −3.92640288695859753922105177849, −2.69438071795037978955720264563, −2.344840848373414845743474511836, −0.48896517146324773894302407738, 0.69670621887289507596587811279, 1.99235113182238301917001216329, 2.74784894080394604121983647362, 4.50334262583648213209128816893, 5.21825327306192474571133902125, 6.21975106647876979267808900795, 6.92842572047666450819513880287, 7.51578006784354873195603245929, 8.52764523914011289772139298581, 8.87413962080150739948757507740, 10.26613166165817359114425581542, 10.75217230865925955545144019023, 11.96152953759692552405224638331, 12.82926561886264359227674787167, 13.49626425541219588314088734777, 14.306560926160189900779378026615, 15.05271547837054723434849499363, 15.82706307501863779567502405471, 16.82629664415566719273786793448, 17.41376347086381974557579751029, 18.026687414948821171382312053606, 18.845266943962867718215313953819, 19.365951652286978939343658877220, 20.06700270699626756055931980843, 21.1331528348784434856858528739

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