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Properties

Label 1-65e2-4225.113-r1-0-0
Degree $1$
Conductor $4225$
Sign $0.0978 - 0.995i$
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  + (0.561 + 0.827i)2-s + (0.857 + 0.513i)3-s + (−0.369 + 0.929i)4-s + (0.0563 + 0.998i)6-s + (−0.316 − 0.948i)7-s + (−0.976 + 0.215i)8-s + (0.471 + 0.881i)9-s + (−0.471 + 0.881i)11-s + (−0.794 + 0.607i)12-s + (0.607 − 0.794i)14-s + (−0.726 − 0.686i)16-s + (−0.301 + 0.953i)17-s + (−0.464 + 0.885i)18-s + (−0.913 + 0.406i)19-s + (0.215 − 0.976i)21-s + (−0.994 + 0.104i)22-s + ⋯
L(s)  = 1  + (0.561 + 0.827i)2-s + (0.857 + 0.513i)3-s + (−0.369 + 0.929i)4-s + (0.0563 + 0.998i)6-s + (−0.316 − 0.948i)7-s + (−0.976 + 0.215i)8-s + (0.471 + 0.881i)9-s + (−0.471 + 0.881i)11-s + (−0.794 + 0.607i)12-s + (0.607 − 0.794i)14-s + (−0.726 − 0.686i)16-s + (−0.301 + 0.953i)17-s + (−0.464 + 0.885i)18-s + (−0.913 + 0.406i)19-s + (0.215 − 0.976i)21-s + (−0.994 + 0.104i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0978 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0978 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.0978 - 0.995i$
Analytic conductor: \(454.039\)
Root analytic conductor: \(454.039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (1:\ ),\ 0.0978 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.378200181 + 1.249312563i\)
\(L(\frac12)\) \(\approx\) \(-1.378200181 + 1.249312563i\)
\(L(1)\) \(\approx\) \(0.9491993434 + 1.128192391i\)
\(L(1)\) \(\approx\) \(0.9491993434 + 1.128192391i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.561 + 0.827i)T \)
3 \( 1 + (0.857 + 0.513i)T \)
7 \( 1 + (-0.316 - 0.948i)T \)
11 \( 1 + (-0.471 + 0.881i)T \)
17 \( 1 + (-0.301 + 0.953i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (0.789 + 0.613i)T \)
31 \( 1 + (0.836 - 0.548i)T \)
37 \( 1 + (-0.988 - 0.152i)T \)
41 \( 1 + (0.0884 + 0.996i)T \)
43 \( 1 + (-0.160 + 0.987i)T \)
47 \( 1 + (0.896 + 0.443i)T \)
53 \( 1 + (-0.0965 + 0.995i)T \)
59 \( 1 + (-0.991 + 0.128i)T \)
61 \( 1 + (0.870 - 0.493i)T \)
67 \( 1 + (0.929 - 0.369i)T \)
71 \( 1 + (-0.999 - 0.0161i)T \)
73 \( 1 + (0.849 - 0.527i)T \)
79 \( 1 + (0.443 - 0.896i)T \)
83 \( 1 + (0.873 + 0.485i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (-0.650 - 0.759i)T \)
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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

−18.07038226990248112693333001075, −17.238923079368609259689412410171, −15.882744630214644499046235453141, −15.54644378668543676983023492478, −14.87347568924918332876281153038, −13.99287019680682487427733961761, −13.64251028099016343806420095734, −12.92356209389887152864462550057, −12.302402578635289337875367156669, −11.77794553721121076006395296939, −10.89163640370094176176158202998, −10.16662898173978330853025346924, −9.35397530261803299748718979496, −8.576527981967254175871732716285, −8.471096792645751316130679794784, −6.96999501753691813709188828957, −6.50742913364422746954878701541, −5.58606086339982030050570200367, −4.88889195696903902185661134017, −3.939525416785279068556368695004, −3.08229895874872009001849975151, −2.58393669981500585106624356665, −2.07775810985156278271135063356, −0.86347924173060306813483916059, −0.20520358761813946432437066200, 1.3708973099917392837200035285, 2.4541642483107848699818500674, 3.21713538195448157803656887463, 3.985681790998686058543162943938, 4.4951957956598794968298685290, 5.12767118241375387240089362806, 6.28401593909100750551275825106, 6.86108321439794615658945893030, 7.731483165542267708009268042959, 8.068869085842758210356124545335, 8.96949687316671975492678548273, 9.66039882718355599113288507964, 10.41700340646871600196821525990, 11.00243306797273013639317704611, 12.32707941753131801496586870603, 12.84710646901058967882024555314, 13.46070340602172931217700294483, 14.02186585056965427654923279425, 14.81103857980173300636674552817, 15.23364404074335012794763564105, 15.84397127043550760798442127418, 16.58560447343994723424217969863, 17.21442780263960116211955721342, 17.73057341714757626855675180775, 18.815804042032946116750313208396

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