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Properties

Label 1-605-605.218-r1-0-0
Degree $1$
Conductor $605$
Sign $-0.350 + 0.936i$
Analytic cond. $65.0162$
Root an. cond. $65.0162$
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  + (0.884 + 0.466i)2-s + (0.951 + 0.309i)3-s + (0.564 + 0.825i)4-s + (0.696 + 0.717i)6-s + (0.967 + 0.254i)7-s + (0.113 + 0.993i)8-s + (0.809 + 0.587i)9-s + (0.281 + 0.959i)12-s + (−0.336 + 0.941i)13-s + (0.736 + 0.676i)14-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (0.441 + 0.897i)18-s + (−0.516 − 0.856i)19-s + (0.841 + 0.540i)21-s + ⋯
L(s)  = 1  + (0.884 + 0.466i)2-s + (0.951 + 0.309i)3-s + (0.564 + 0.825i)4-s + (0.696 + 0.717i)6-s + (0.967 + 0.254i)7-s + (0.113 + 0.993i)8-s + (0.809 + 0.587i)9-s + (0.281 + 0.959i)12-s + (−0.336 + 0.941i)13-s + (0.736 + 0.676i)14-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (0.441 + 0.897i)18-s + (−0.516 − 0.856i)19-s + (0.841 + 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(65.0162\)
Root analytic conductor: \(65.0162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (1:\ ),\ -0.350 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.472924886 + 5.005958162i\)
\(L(\frac12)\) \(\approx\) \(3.472924886 + 5.005958162i\)
\(L(1)\) \(\approx\) \(2.320762478 + 1.493731700i\)
\(L(1)\) \(\approx\) \(2.320762478 + 1.493731700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.884 + 0.466i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (0.967 + 0.254i)T \)
13 \( 1 + (-0.336 + 0.941i)T \)
17 \( 1 + (0.226 - 0.974i)T \)
19 \( 1 + (-0.516 - 0.856i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (0.610 + 0.791i)T \)
37 \( 1 + (-0.999 + 0.0285i)T \)
41 \( 1 + (-0.985 - 0.170i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.441 - 0.897i)T \)
53 \( 1 + (0.931 - 0.362i)T \)
59 \( 1 + (0.985 - 0.170i)T \)
61 \( 1 + (-0.466 - 0.884i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.633 + 0.774i)T \)
79 \( 1 + (0.870 - 0.491i)T \)
83 \( 1 + (-0.0570 - 0.998i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.980 - 0.198i)T \)
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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

−22.63000439655052103004480970040, −21.54950646838354495080985712454, −20.8619817541871497834508836641, −20.398227954623139626242236945311, −19.451978263470817005853348879009, −18.823519542869571770005326098972, −17.80488615568993647936719536929, −16.73286473166571973894068470481, −15.260739994865892275141493651521, −14.9878322196364381092397429838, −14.15118069092951584235858754155, −13.39666856371365188704714083770, −12.540992014907229219441018195040, −11.866172483104913420423188529171, −10.495956390074111292783625435252, −10.15513632138047643322327635415, −8.62396554638867313577912228849, −7.92930262274921587192487220185, −6.893103528183464485519473284908, −5.81626932935633299160235185969, −4.6618491459613150607660946107, −3.86419452697996506363988636589, −2.82054537616962522423050372569, −1.892397141376898631179686454147, −0.93684748066500339421908320774, 1.67637690947962068007770478674, 2.60340367325834027575446166030, 3.56604365292183295754682575582, 4.7915628577601224508154627044, 5.036755694400117051628849827698, 6.75070607698296970124059777802, 7.364675381240266820939808649020, 8.455494640197238147553996501264, 9.0115959897642233484596778541, 10.35595400796212615114927667039, 11.48493591881551925724825344731, 12.13733702151501045761620021132, 13.44172801535961705262492536528, 13.91015847027258854581786774223, 14.71014759113425266303052592288, 15.358201356155911018904204453627, 16.13707843474837387445965023820, 17.11454058373798633761366352517, 18.05307161745239080682369925568, 19.20024567524111968922186576871, 20.03846796873236555750665262900, 20.96207284694935591974683214257, 21.43354986609553691553884039743, 22.04384892142820941693654964489, 23.32587116399617252768530656421

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