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Properties

Label 1-605-605.114-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.502 + 0.864i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (0.921 + 0.389i)2-s + (0.809 + 0.587i)3-s + (0.696 + 0.717i)4-s + (0.516 + 0.856i)6-s + (−0.941 + 0.336i)7-s + (0.362 + 0.931i)8-s + (0.309 + 0.951i)9-s + (0.142 + 0.989i)12-s + (−0.897 + 0.441i)13-s + (−0.998 − 0.0570i)14-s + (−0.0285 + 0.999i)16-s + (0.736 + 0.676i)17-s + (−0.0855 + 0.996i)18-s + (0.198 − 0.980i)19-s + (−0.959 − 0.281i)21-s + ⋯
L(s)  = 1  + (0.921 + 0.389i)2-s + (0.809 + 0.587i)3-s + (0.696 + 0.717i)4-s + (0.516 + 0.856i)6-s + (−0.941 + 0.336i)7-s + (0.362 + 0.931i)8-s + (0.309 + 0.951i)9-s + (0.142 + 0.989i)12-s + (−0.897 + 0.441i)13-s + (−0.998 − 0.0570i)14-s + (−0.0285 + 0.999i)16-s + (0.736 + 0.676i)17-s + (−0.0855 + 0.996i)18-s + (0.198 − 0.980i)19-s + (−0.959 − 0.281i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.502 + 0.864i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.502 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.407632883 + 2.447866904i\)
\(L(\frac12)\) \(\approx\) \(1.407632883 + 2.447866904i\)
\(L(1)\) \(\approx\) \(1.669904756 + 1.200618365i\)
\(L(1)\) \(\approx\) \(1.669904756 + 1.200618365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.921 + 0.389i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.941 + 0.336i)T \)
13 \( 1 + (-0.897 + 0.441i)T \)
17 \( 1 + (0.736 + 0.676i)T \)
19 \( 1 + (0.198 - 0.980i)T \)
23 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.870 - 0.491i)T \)
31 \( 1 + (-0.985 + 0.170i)T \)
37 \( 1 + (0.466 + 0.884i)T \)
41 \( 1 + (0.974 - 0.226i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.0855 - 0.996i)T \)
53 \( 1 + (0.0285 + 0.999i)T \)
59 \( 1 + (0.974 + 0.226i)T \)
61 \( 1 + (-0.921 + 0.389i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.610 + 0.791i)T \)
79 \( 1 + (0.774 + 0.633i)T \)
83 \( 1 + (0.564 - 0.825i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.254 - 0.967i)T \)
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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

−22.8939780584553758246375290740, −22.04847296650763676954011636949, −20.980267705083863367982192065350, −20.360297725181377319365283669005, −19.58124884719888592927689287767, −19.03785549730422042278439764459, −18.14857625398923316194916585906, −16.719998140293756171898218782546, −15.96611302793721259983785153836, −14.766759794651421636347009175724, −14.45287301969979643921491900106, −13.35512121636113146935039841434, −12.77210316824996808968553835996, −12.17989218240329405934682042414, −10.996286625150261687584699314966, −9.804163933327358629515511608191, −9.38423183260741317709198726074, −7.69017131067665691921113237744, −7.17444431530010854350454765380, −6.1316027381557842004918006770, −5.13767206441570588103362326392, −3.73741542866613925404706332492, −3.17856338098947910713849574344, −2.21566360723349501416219547077, −0.94456698730544103072882516724, 2.11041160232057413893341168739, 2.97069254079192883828888812599, 3.74916139861111850956327198471, 4.77160044431536968372055193163, 5.64146407803864485568711478017, 6.84686915661232304834532021149, 7.57626088260919031576683131445, 8.73828236088711768696540247808, 9.517822224514207996200163241127, 10.535726154528401229537808360418, 11.635020281615623943392375859787, 12.706706479071756318850930712625, 13.263975871279590368461707520263, 14.23715480070630882102010936839, 15.04204365827096169520604876446, 15.48982593443516208036789584064, 16.61455923476586665438736471476, 16.939000251747807893447788804388, 18.6181958596796362090998432025, 19.5236044535431113804889649442, 20.10119705759785288350153903750, 21.1337196663722585095722101900, 21.802508610264694357459909298963, 22.30984828204796324001579432546, 23.27659754051914502386176861549

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