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Properties

Label 1-605-605.544-r0-0-0
Degree $1$
Conductor $605$
Sign $0.318 - 0.947i$
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  + (0.998 − 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (−0.362 − 0.931i)6-s + (0.921 − 0.389i)7-s + (0.985 − 0.170i)8-s + (−0.809 + 0.587i)9-s + (−0.415 − 0.909i)12-s + (0.870 + 0.491i)13-s + (0.897 − 0.441i)14-s + (0.974 − 0.226i)16-s + (−0.941 − 0.336i)17-s + (−0.774 + 0.633i)18-s + (−0.0285 − 0.999i)19-s + (−0.654 − 0.755i)21-s + ⋯
L(s)  = 1  + (0.998 − 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (−0.362 − 0.931i)6-s + (0.921 − 0.389i)7-s + (0.985 − 0.170i)8-s + (−0.809 + 0.587i)9-s + (−0.415 − 0.909i)12-s + (0.870 + 0.491i)13-s + (0.897 − 0.441i)14-s + (0.974 − 0.226i)16-s + (−0.941 − 0.336i)17-s + (−0.774 + 0.633i)18-s + (−0.0285 − 0.999i)19-s + (−0.654 − 0.755i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.318 - 0.947i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (544, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.318 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.203288703 - 1.583368119i\)
\(L(\frac12)\) \(\approx\) \(2.203288703 - 1.583368119i\)
\(L(1)\) \(\approx\) \(1.798577883 - 0.7348030874i\)
\(L(1)\) \(\approx\) \(1.798577883 - 0.7348030874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.998 - 0.0570i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.921 - 0.389i)T \)
13 \( 1 + (0.870 + 0.491i)T \)
17 \( 1 + (-0.941 - 0.336i)T \)
19 \( 1 + (-0.0285 - 0.999i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.564 + 0.825i)T \)
31 \( 1 + (0.198 + 0.980i)T \)
37 \( 1 + (0.736 + 0.676i)T \)
41 \( 1 + (-0.254 + 0.967i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.774 - 0.633i)T \)
53 \( 1 + (-0.974 - 0.226i)T \)
59 \( 1 + (-0.254 - 0.967i)T \)
61 \( 1 + (-0.998 - 0.0570i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (-0.516 - 0.856i)T \)
79 \( 1 + (0.696 + 0.717i)T \)
83 \( 1 + (-0.0855 - 0.996i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.466 + 0.884i)T \)
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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

−23.01988865745385055211504249985, −22.46017424934393045548852064342, −21.557381681318093063481001770651, −20.91195788447571601764484726957, −20.486995598468958730653870703650, −19.3550207966165932034458092316, −18.0448540066968141496229615729, −17.21395875060266503248911000776, −16.36035919924658155732040769104, −15.37099759455577159547163275255, −15.08742449599433183183196818901, −14.1198818945852065962184368833, −13.19013302525476369008464224935, −12.15483701211836208484750784602, −11.1383823139821471574413774428, −10.9703230175587302968893788588, −9.675260847409082349503487149310, −8.50763185020133580441644746566, −7.62814574307515246577675336745, −6.085505939899442178883656479964, −5.694781952539584379168248194273, −4.58511390540880835584004546472, −3.94512580010743862802075999740, −2.855714468378058317177166965521, −1.58882032041526123490936840370, 1.16718243905723184488998341646, 2.04971810733794758661289754754, 3.16661509634409095810410500210, 4.56722740460361613469720465418, 5.13576273531510715357716299811, 6.47050206343625604742043937601, 6.885138858308430432803147243454, 7.94593891220916519543151943722, 8.897404123167753879759799842337, 10.74078501665132802265567093795, 11.15727878905768730875350606585, 11.892581392598701403636575764905, 12.972165695697142968645526519319, 13.53748017613889753197744495197, 14.25832029273083716945564150189, 15.13996268303890652916011962517, 16.23363289856163251050294294260, 17.04630026997183470027072290316, 17.96498830712875522698725574932, 18.75082596526937936721118464982, 19.88638650060217611223710542557, 20.37997160738871841843765956240, 21.3948212883100239445908117362, 22.189326763484081517953618167582, 23.113034668511267537570903450826

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