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Properties

Label 1-605-605.428-r0-0-0
Degree $1$
Conductor $605$
Sign $0.973 - 0.227i$
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  + (−0.540 + 0.841i)2-s + i·3-s + (−0.415 − 0.909i)4-s + (−0.841 − 0.540i)6-s + (0.755 − 0.654i)7-s + (0.989 + 0.142i)8-s − 9-s + (0.909 − 0.415i)12-s + (0.909 − 0.415i)13-s + (0.142 + 0.989i)14-s + (−0.654 + 0.755i)16-s + (−0.281 − 0.959i)17-s + (0.540 − 0.841i)18-s + (−0.959 − 0.281i)19-s + (0.654 + 0.755i)21-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + i·3-s + (−0.415 − 0.909i)4-s + (−0.841 − 0.540i)6-s + (0.755 − 0.654i)7-s + (0.989 + 0.142i)8-s − 9-s + (0.909 − 0.415i)12-s + (0.909 − 0.415i)13-s + (0.142 + 0.989i)14-s + (−0.654 + 0.755i)16-s + (−0.281 − 0.959i)17-s + (0.540 − 0.841i)18-s + (−0.959 − 0.281i)19-s + (0.654 + 0.755i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7544446544 - 0.08677474573i\)
\(L(\frac12)\) \(\approx\) \(0.7544446544 - 0.08677474573i\)
\(L(1)\) \(\approx\) \(0.7076300805 + 0.2516201397i\)
\(L(1)\) \(\approx\) \(0.7076300805 + 0.2516201397i\)

Euler product

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

−23.17123889070054270048455246774, −22.06434352005537862873104780669, −21.32583988881817998662941240237, −20.55390267799565592426194109096, −19.61505840274932925488777407386, −18.900312763001972835064438613173, −18.30963729930371429422503294198, −17.54555022321832378795445486402, −16.90914894023980530509640430597, −15.63744130297062377120431538622, −14.44006492024662815436196752866, −13.61470463592468306971969733249, −12.73497268193584912142043948569, −12.05166109663329635133161550867, −11.25301181438679800565067330740, −10.56845775958209997365402969088, −9.12204460472320927851546129223, −8.45379785815081601104462800933, −7.86712083115563912999552601560, −6.651197400837774364389940311475, −5.648904265175523014301363952094, −4.28686326801942851378246363457, −3.1527807230953861433916541798, −1.871353902875110474770511874353, −1.52149339943485750333403082896, 0.46428657995535438993921010571, 2.10161080357184012751370328647, 3.810713854316425428115946418802, 4.57194586254939017717740754722, 5.4727173053175654113636613917, 6.42513791867915225462129712502, 7.592183860175852750117095585886, 8.44593221927576330763114060162, 9.132234216248889340341175242730, 10.24242950695764432669555954473, 10.7658358669036671988775448364, 11.615374594036671946971417330774, 13.365763016131365629922389590181, 14.03393224421943667154096916559, 14.914267111640580506300718053826, 15.55559421742753448838081628368, 16.40422348269288640912440362398, 17.08755741869253986083447876107, 17.83297876751167150678087136370, 18.674927539187346093814882938202, 19.88569207249627192810602171619, 20.512765944967029814825671297796, 21.26234413104978595397499594165, 22.58671448798556742848718908224, 22.9310687279946197346314208265

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