↑

Properties

Label 1-605-605.318-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.913 - 0.407i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.989 − 0.142i)2-s + i·3-s + (0.959 + 0.281i)4-s + (0.142 − 0.989i)6-s + (0.540 + 0.841i)7-s + (−0.909 − 0.415i)8-s − 9-s + (−0.281 + 0.959i)12-s + (−0.281 + 0.959i)13-s + (−0.415 − 0.909i)14-s + (0.841 + 0.540i)16-s + (−0.755 − 0.654i)17-s + (0.989 + 0.142i)18-s + (−0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + i·3-s + (0.959 + 0.281i)4-s + (0.142 − 0.989i)6-s + (0.540 + 0.841i)7-s + (−0.909 − 0.415i)8-s − 9-s + (−0.281 + 0.959i)12-s + (−0.281 + 0.959i)13-s + (−0.415 − 0.909i)14-s + (0.841 + 0.540i)16-s + (−0.755 − 0.654i)17-s + (0.989 + 0.142i)18-s + (−0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.913 - 0.407i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.913 - 0.407i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06316115321 + 0.2962896619i\)
\(L(\frac12)\) \(\approx\) \(-0.06316115321 + 0.2962896619i\)
\(L(1)\) \(\approx\) \(0.4918296899 + 0.2525262143i\)
\(L(1)\) \(\approx\) \(0.4918296899 + 0.2525262143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.989 - 0.142i)T \)
3 \( 1 + iT \)
7 \( 1 + (0.540 + 0.841i)T \)
13 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-0.540 + 0.841i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (0.281 + 0.959i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 + (-0.989 + 0.142i)T \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (0.989 + 0.142i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (-0.540 - 0.841i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.909 - 0.415i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.95376186429528473718625687185, −21.670693087222229612516564374909, −20.347588897696872138010299507399, −20.12282547782970254870855616495, −19.23911498834168794024920001559, −18.24395727170064930376457467601, −17.7946324804975817634748821475, −16.97545384714214523898194386343, −16.32613052272379987217851140536, −14.83510301354276787552856616900, −14.50399367700860254020570545286, −13.099735910416973827253194171918, −12.490109878860672301042450172561, −11.24182250713691308884811617695, −10.758162985762006336412568046475, −9.7240537735813830640358120468, −8.3436061375140447394693369447, −8.07183706991495044509470807134, −7.06744317299278506004275952754, −6.329254355285267906591287127157, −5.27782521420990518467867123970, −3.65447797997406594054497195529, −2.27739441072368771552679041031, −1.490923726600208588549636317674, −0.19892479831994204618828622433, 1.888786673012469874990209979550, 2.68331516865913314811540467133, 3.97842615923599829071688846078, 5.06172467412798030160851319006, 6.10430442084029376800576058292, 7.21322940414156882922487209522, 8.38699333848948265371979931883, 9.0839526712820027511136630006, 9.61507824425413075242597429711, 10.744936164132884378182690647296, 11.49798777487704592047675284326, 11.97905835617145343059909188609, 13.50789249269082758016888584967, 14.75994969794192780893358695946, 15.33262075145628404101584723426, 16.08812262519284298473140511191, 16.93748237507681263464618933219, 17.681260435950583157256366594600, 18.51306662623536942845682471582, 19.46352775855045015072280960104, 20.19729522010685105248085468383, 21.10909736041725487446875246685, 21.64547967230802723204850794819, 22.32058793415934321658452916194, 23.77511661836991744450066813148

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