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Properties

Label 1-605-605.389-r0-0-0
Degree $1$
Conductor $605$
Sign $0.338 + 0.940i$
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  + (−0.897 + 0.441i)2-s + (0.809 + 0.587i)3-s + (0.610 − 0.791i)4-s + (−0.985 − 0.170i)6-s + (0.998 − 0.0570i)7-s + (−0.198 + 0.980i)8-s + (0.309 + 0.951i)9-s + (0.959 − 0.281i)12-s + (0.564 + 0.825i)13-s + (−0.870 + 0.491i)14-s + (−0.254 − 0.967i)16-s + (0.921 − 0.389i)17-s + (−0.696 − 0.717i)18-s + (0.974 − 0.226i)19-s + (0.841 + 0.540i)21-s + ⋯
L(s)  = 1  + (−0.897 + 0.441i)2-s + (0.809 + 0.587i)3-s + (0.610 − 0.791i)4-s + (−0.985 − 0.170i)6-s + (0.998 − 0.0570i)7-s + (−0.198 + 0.980i)8-s + (0.309 + 0.951i)9-s + (0.959 − 0.281i)12-s + (0.564 + 0.825i)13-s + (−0.870 + 0.491i)14-s + (−0.254 − 0.967i)16-s + (0.921 − 0.389i)17-s + (−0.696 − 0.717i)18-s + (0.974 − 0.226i)19-s + (0.841 + 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.338 + 0.940i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.338 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.209988329 + 0.8506287286i\)
\(L(\frac12)\) \(\approx\) \(1.209988329 + 0.8506287286i\)
\(L(1)\) \(\approx\) \(1.021487779 + 0.4313256742i\)
\(L(1)\) \(\approx\) \(1.021487779 + 0.4313256742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.897 + 0.441i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.998 - 0.0570i)T \)
13 \( 1 + (0.564 + 0.825i)T \)
17 \( 1 + (0.921 - 0.389i)T \)
19 \( 1 + (0.974 - 0.226i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (0.0855 - 0.996i)T \)
31 \( 1 + (-0.0285 - 0.999i)T \)
37 \( 1 + (-0.941 + 0.336i)T \)
41 \( 1 + (-0.466 + 0.884i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.696 + 0.717i)T \)
53 \( 1 + (0.254 - 0.967i)T \)
59 \( 1 + (-0.466 - 0.884i)T \)
61 \( 1 + (0.897 + 0.441i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (0.516 + 0.856i)T \)
73 \( 1 + (0.362 - 0.931i)T \)
79 \( 1 + (0.993 + 0.113i)T \)
83 \( 1 + (-0.774 + 0.633i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.736 - 0.676i)T \)
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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

−22.98565408579991117302428663283, −21.67267438928708311297833124158, −20.96010035130538916695634060466, −20.272993840542823091190300387199, −19.72255679643723577754737252143, −18.56313891684555408416594758958, −18.188279877048802859483494014782, −17.48619277858432905053108280438, −16.33182508297564325881775723816, −15.41105453916069752570184281212, −14.45880521290609923240027784327, −13.65560571153065371000164449972, −12.45365642047350483539597605530, −12.024896576847527981844231172180, −10.81140047004515471275837803790, −10.06488426198107049478276771300, −8.93830653933091038810393222901, −8.215176553564969253558869057268, −7.70714451495996000208820342410, −6.707718651595564999369314288363, −5.402555861965907398380007275408, −3.769472444505629407703373427546, −3.02288222408391092292972573153, −1.79839165413187281169611217221, −1.10378944037636276542339337596, 1.35701489006740027998093616236, 2.25661466169964005044177842900, 3.59796625038933031460132900850, 4.798240233509256320439899167920, 5.68069133451941329437321741507, 7.05618189265535386501802061585, 7.93556575953733724855612747700, 8.453508628866183825368187867, 9.54875793668902880713435071922, 10.01643050434269307852890709233, 11.24240594904525639789967497460, 11.74846923920110471468649511744, 13.65871897036843644812608165072, 14.13417300313761828385660049105, 14.98699839545958657797470390547, 15.78236417312338332148132167084, 16.48274121796544395005664798490, 17.37972193587181754517775318334, 18.371638604894329796891522556566, 18.97168951058334311003363889370, 19.93837653042916194291556559679, 20.7440136863710346426025404908, 21.1418217021131799877490487435, 22.33677916702500956361584793835, 23.58533247608914827257385369519

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