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Properties

Label 1-605-605.434-r0-0-0
Degree $1$
Conductor $605$
Sign $0.0337 + 0.999i$
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  + (−0.610 + 0.791i)2-s + (−0.309 − 0.951i)3-s + (−0.254 − 0.967i)4-s + (0.941 + 0.336i)6-s + (−0.993 + 0.113i)7-s + (0.921 + 0.389i)8-s + (−0.809 + 0.587i)9-s + (−0.841 + 0.540i)12-s + (0.362 − 0.931i)13-s + (0.516 − 0.856i)14-s + (−0.870 + 0.491i)16-s + (−0.696 + 0.717i)17-s + (0.0285 − 0.999i)18-s + (0.897 − 0.441i)19-s + (0.415 + 0.909i)21-s + ⋯
L(s)  = 1  + (−0.610 + 0.791i)2-s + (−0.309 − 0.951i)3-s + (−0.254 − 0.967i)4-s + (0.941 + 0.336i)6-s + (−0.993 + 0.113i)7-s + (0.921 + 0.389i)8-s + (−0.809 + 0.587i)9-s + (−0.841 + 0.540i)12-s + (0.362 − 0.931i)13-s + (0.516 − 0.856i)14-s + (−0.870 + 0.491i)16-s + (−0.696 + 0.717i)17-s + (0.0285 − 0.999i)18-s + (0.897 − 0.441i)19-s + (0.415 + 0.909i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0337 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0337 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.0337 + 0.999i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (434, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.0337 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3047797180 + 0.2946623783i\)
\(L(\frac12)\) \(\approx\) \(0.3047797180 + 0.2946623783i\)
\(L(1)\) \(\approx\) \(0.5406201992 + 0.06379660048i\)
\(L(1)\) \(\approx\) \(0.5406201992 + 0.06379660048i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.610 + 0.791i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.993 + 0.113i)T \)
13 \( 1 + (0.362 - 0.931i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (0.897 - 0.441i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.985 - 0.170i)T \)
31 \( 1 + (-0.998 + 0.0570i)T \)
37 \( 1 + (-0.774 + 0.633i)T \)
41 \( 1 + (-0.564 - 0.825i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.0285 + 0.999i)T \)
53 \( 1 + (0.870 + 0.491i)T \)
59 \( 1 + (-0.564 + 0.825i)T \)
61 \( 1 + (0.610 + 0.791i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (0.736 + 0.676i)T \)
79 \( 1 + (0.974 + 0.226i)T \)
83 \( 1 + (-0.198 + 0.980i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.0855 + 0.996i)T \)
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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

−22.477250735986685382728144496028, −22.08380894198151139659358627690, −21.123432171682390489151268342283, −20.34346921186307373832937313942, −19.80572369261202105816114673921, −18.66630599693270414266786809280, −18.07242928729935609893880342781, −16.86942972216659936095030959006, −16.34306484153076134060592888507, −15.791634832408586924613000130903, −14.39600674960567202441506999653, −13.47209403819430301458158098774, −12.45490003745811797792309955382, −11.58365215840257815787210524933, −10.8927392328357337106693884246, −9.94178508668128030889644190892, −9.347008457398623466331753055569, −8.67954273854653341432922390417, −7.27796525475676214354292845655, −6.304930401031357195280756625251, −4.98639318126238215485693716870, −3.90887106996694784023643623404, −3.30289185737784064797480567723, −2.0613171227918297181881684812, −0.32884106376398803294058424482, 1.01775351336789853482256270638, 2.25631383971140043062294610469, 3.65758655585130254290706216865, 5.41070568064671899862995419853, 5.856089607169502879927199375319, 6.885218127616721183423972854110, 7.49637713412214591941080609284, 8.505673308377074307236280015854, 9.33492499661885354623982143077, 10.4066508302868530537757964847, 11.23955301473427968272609302112, 12.45050539401750949892463031443, 13.309671860735803930629243306545, 13.861174542992719258503626471471, 15.20325893931589734039344291974, 15.792615950543232428233994524088, 16.77378973740374213821100611379, 17.515010852659313302503342928764, 18.2015485771211731722709129559, 18.966985055797387449030669246704, 19.71597498943981388517021400467, 20.31745564753139542768033117864, 22.22731603357478078661481438416, 22.50334578941684887497191982511, 23.54162770732801279890037917480

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