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Properties

Label 1-605-605.538-r0-0-0
Degree $1$
Conductor $605$
Sign $0.638 - 0.769i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.638 - 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1871971747 - 0.08791380632i\)
\(L(\frac12)\) \(\approx\) \(0.1871971747 - 0.08791380632i\)
\(L(1)\) \(\approx\) \(0.6617763062 + 0.5130450038i\)
\(L(1)\) \(\approx\) \(0.6617763062 + 0.5130450038i\)

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

−23.24811398802465791672185240757, −22.21552846893556926241881588483, −21.75866764719412328715354245833, −20.656207472895953884020933011116, −19.6889100925096737008742170223, −19.03539683345148656792282882889, −18.767077353767337436414060305684, −17.50246174904512699011845888904, −16.77076722402267541591845468344, −15.14252022886672191374095352710, −14.814857349041249966620942197115, −13.49674036533134577152208200793, −13.03650549846857261986446374007, −12.22379595631961041190408122209, −11.63329407872864285310020287348, −10.568132653233565222123106175424, −9.47494338101844025892973935763, −8.75564713996179740181086969956, −7.4754920923441721944750260083, −6.325594071387707217961707168016, −5.78515111103053127717204222171, −4.56017190471006328377947082596, −3.27054699769640913078619014289, −2.43204995254607761376795020118, −1.556608638623819591998622836375, 0.08202744848045604878994234067, 2.7628510037055593933214353736, 3.48403494845130234369174215723, 4.53620854779309529319787054639, 5.182015312115077683001070709877, 6.3076754978378077374475486459, 7.16195443634123291826104632458, 8.1996484679203272249722101416, 9.23081638711384402448944490229, 9.957448878265417771894466928174, 10.94508234438424702290869627774, 12.130108497253951973159531873921, 12.975221280682007399178682656613, 13.91857077459141725695132601811, 14.752114132341698846703786796406, 15.33166585549041346456350690719, 16.45515610749879764622069044556, 16.67785523731169749286179296492, 17.54684393302655053571549212918, 18.75425060749800543426149312043, 19.95234666969548884540651232377, 20.61249337394314125454913240161, 21.58388338978722711192532279040, 22.222112268079375599830087128, 23.04293145243803494941348212582

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