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Properties

Label 1-65e2-4225.42-r1-0-0
Degree $1$
Conductor $4225$
Sign $0.832 + 0.553i$
Analytic cond. $454.039$
Root an. cond. $454.039$
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  + (0.650 + 0.759i)2-s + (0.968 − 0.247i)3-s + (−0.152 + 0.988i)4-s + (0.818 + 0.574i)6-s + (−0.721 + 0.692i)7-s + (−0.849 + 0.527i)8-s + (0.877 − 0.478i)9-s + (−0.877 − 0.478i)11-s + (0.0965 + 0.995i)12-s + (−0.995 − 0.0965i)14-s + (−0.953 − 0.301i)16-s + (−0.881 − 0.471i)17-s + (0.935 + 0.354i)18-s + (−0.669 + 0.743i)19-s + (−0.527 + 0.849i)21-s + (−0.207 − 0.978i)22-s + ⋯
L(s)  = 1  + (0.650 + 0.759i)2-s + (0.968 − 0.247i)3-s + (−0.152 + 0.988i)4-s + (0.818 + 0.574i)6-s + (−0.721 + 0.692i)7-s + (−0.849 + 0.527i)8-s + (0.877 − 0.478i)9-s + (−0.877 − 0.478i)11-s + (0.0965 + 0.995i)12-s + (−0.995 − 0.0965i)14-s + (−0.953 − 0.301i)16-s + (−0.881 − 0.471i)17-s + (0.935 + 0.354i)18-s + (−0.669 + 0.743i)19-s + (−0.527 + 0.849i)21-s + (−0.207 − 0.978i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(454.039\)
Root analytic conductor: \(454.039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (1:\ ),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.266594469 + 0.6850915407i\)
\(L(\frac12)\) \(\approx\) \(2.266594469 + 0.6850915407i\)
\(L(1)\) \(\approx\) \(1.325649963 + 0.6199190902i\)
\(L(1)\) \(\approx\) \(1.325649963 + 0.6199190902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.650 + 0.759i)T \)
3 \( 1 + (0.968 - 0.247i)T \)
7 \( 1 + (-0.721 + 0.692i)T \)
11 \( 1 + (-0.877 - 0.478i)T \)
17 \( 1 + (-0.881 - 0.471i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.994 + 0.104i)T \)
29 \( 1 + (0.231 - 0.972i)T \)
31 \( 1 + (-0.906 + 0.421i)T \)
37 \( 1 + (-0.857 + 0.513i)T \)
41 \( 1 + (0.997 + 0.0643i)T \)
43 \( 1 + (0.391 + 0.919i)T \)
47 \( 1 + (-0.0483 - 0.998i)T \)
53 \( 1 + (0.997 + 0.0724i)T \)
59 \( 1 + (0.581 + 0.813i)T \)
61 \( 1 + (-0.789 - 0.613i)T \)
67 \( 1 + (-0.988 + 0.152i)T \)
71 \( 1 + (-0.962 - 0.270i)T \)
73 \( 1 + (0.999 - 0.0241i)T \)
79 \( 1 + (0.998 - 0.0483i)T \)
83 \( 1 + (0.698 - 0.715i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (-0.493 + 0.870i)T \)
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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

−18.25879362173162382205255559686, −17.766848086270488629427420964527, −16.57538471966634258734000108432, −15.74394463577855487651534635642, −15.436256612740486458623858266686, −14.57996973466470805220152559513, −13.97941304560663824439127313194, −13.30551059846509458863586217735, −12.84009790208135080367329379135, −12.35579358267833785632783091621, −11.043000268933702333110516710468, −10.55254170704369937704915019755, −10.10535633681070752392190105332, −9.2098992870845962831504981878, −8.772033045669533564549392360983, −7.633558356701644316789227904543, −6.99665554211005144386121276204, −6.178279408315117538067135695913, −5.181529961715624677259622566916, −4.33705534689472447406962112964, −3.93395532937339760425376402671, −3.10128642409416602256264008032, −2.34130932289781681921601945443, −1.81607703710243662252029541814, −0.543713223962682288731602704168, 0.33760171217258709713373177200, 2.04009778668832992030142058617, 2.53727330052925823496669976332, 3.32447936206040884737807597266, 3.97646890627940593239444765657, 4.82906365091489693864039932296, 5.81861421139454205640733203040, 6.28816830855158252775023433546, 7.12369820230324264779392422567, 7.84291703447220227482293383828, 8.46974027503371923350672465214, 9.00676387420340959456046961650, 9.78123526361735632601748821373, 10.6690952492428253264443553483, 11.83288866643640533445682189161, 12.37860934389151924507673755510, 13.088156973965424812474426190359, 13.540447520974511822226617771238, 14.133414503928990281832902035380, 15.07084414075909552179430442362, 15.34864542663995459318164375019, 16.18627375473103885873231860182, 16.451465158938313291948441412253, 17.835124065348508067197577956080, 18.12038214035594119114409217579

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