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Properties

Label 1-65e2-4225.714-r0-0-0
Degree $1$
Conductor $4225$
Sign $0.789 + 0.613i$
Analytic cond. $19.6208$
Root an. cond. $19.6208$
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  + (−0.168 − 0.985i)2-s + (−0.926 − 0.377i)3-s + (−0.943 + 0.331i)4-s + (−0.215 + 0.976i)6-s + (0.120 − 0.992i)7-s + (0.485 + 0.873i)8-s + (0.715 + 0.698i)9-s + (−0.715 + 0.698i)11-s + (0.998 + 0.0483i)12-s + (−0.998 + 0.0483i)14-s + (0.779 − 0.626i)16-s + (−0.485 − 0.873i)17-s + (0.568 − 0.822i)18-s + (0.809 − 0.587i)19-s + (−0.485 + 0.873i)21-s + (0.809 + 0.587i)22-s + ⋯
L(s)  = 1  + (−0.168 − 0.985i)2-s + (−0.926 − 0.377i)3-s + (−0.943 + 0.331i)4-s + (−0.215 + 0.976i)6-s + (0.120 − 0.992i)7-s + (0.485 + 0.873i)8-s + (0.715 + 0.698i)9-s + (−0.715 + 0.698i)11-s + (0.998 + 0.0483i)12-s + (−0.998 + 0.0483i)14-s + (0.779 − 0.626i)16-s + (−0.485 − 0.873i)17-s + (0.568 − 0.822i)18-s + (0.809 − 0.587i)19-s + (−0.485 + 0.873i)21-s + (0.809 + 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $0.789 + 0.613i$
Analytic conductor: \(19.6208\)
Root analytic conductor: \(19.6208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4225} (714, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4225,\ (0:\ ),\ 0.789 + 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1945411059 + 0.06674775007i\)
\(L(\frac12)\) \(\approx\) \(0.1945411059 + 0.06674775007i\)
\(L(1)\) \(\approx\) \(0.4656260354 - 0.3247471847i\)
\(L(1)\) \(\approx\) \(0.4656260354 - 0.3247471847i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.168 - 0.985i)T \)
3 \( 1 + (-0.926 - 0.377i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
11 \( 1 + (-0.715 + 0.698i)T \)
17 \( 1 + (-0.485 - 0.873i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.989 - 0.144i)T \)
31 \( 1 + (-0.215 + 0.976i)T \)
37 \( 1 + (-0.861 - 0.506i)T \)
41 \( 1 + (0.527 + 0.849i)T \)
43 \( 1 + (0.748 + 0.663i)T \)
47 \( 1 + (0.0241 - 0.999i)T \)
53 \( 1 + (0.681 - 0.732i)T \)
59 \( 1 + (0.998 + 0.0483i)T \)
61 \( 1 + (0.981 + 0.192i)T \)
67 \( 1 + (-0.943 - 0.331i)T \)
71 \( 1 + (-0.926 - 0.377i)T \)
73 \( 1 + (0.715 - 0.698i)T \)
79 \( 1 + (0.0241 - 0.999i)T \)
83 \( 1 + (0.926 - 0.377i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.262 + 0.964i)T \)
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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

−18.1930812331145382305993185567, −17.53634096414325571494988589482, −16.89931507354225533768872036280, −16.15430711944128320478345252586, −15.797965880656339470883150440053, −15.11711726700254876973622158158, −14.53470974437305697585003293445, −13.614772848557598344490928850322, −12.79138497581125676084637069833, −12.31940368045640033308836753341, −11.36738575746324853607923046866, −10.67469217792352859636691859092, −10.02680294712759896696837236603, −9.21361914641751404803446516920, −8.60132550520211538464023199436, −7.87787042825726848424505441950, −7.04350904320721554393680332150, −6.1674382742036145178192937588, −5.63461238705424398086143188512, −5.33025869264205806225242313628, −4.28450436767378236937743026422, −3.66914619751518446682024768965, −2.40759137640736223229821519590, −1.25120582618441516186231538485, −0.09526010202791936133319156375, 0.83235008027506487681928547949, 1.65470165923428650342998238583, 2.427825980343162730279946481343, 3.47119534098210645054937874092, 4.27538489532290137505501041832, 5.05394105466262303621301336900, 5.41667473198434430838941814730, 6.75960733504147473628869151414, 7.43798037133733390318246742442, 7.791118970469225035900660715811, 9.03203846705784708276844126484, 9.76990820288397333109452561997, 10.34116402009369533585030352112, 11.00581080803524398639187467489, 11.53524158107266528188958624157, 12.14885687819674121856842865615, 13.04685353651770619297599535112, 13.38427049324873267254698051502, 13.99563843063438614811943780671, 15.026351532493156281663706533951, 16.12491633515674474814779484418, 16.44596802025640102753441488650, 17.58273738806324906810640315086, 17.79299717547257816227600273850, 18.17837747345210806731645938491

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