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Properties

Label 1-65-65.4-r0-0-0
Degree $1$
Conductor $65$
Sign $0.859 + 0.511i$
Analytic cond. $0.301858$
Root an. cond. $0.301858$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 12-s + 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + (0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)22-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 12-s + 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + (0.5 − 0.866i)19-s − 21-s + (0.5 − 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ 0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7397294789 + 0.2033022946i\)
\(L(\frac12)\) \(\approx\) \(0.7397294789 + 0.2033022946i\)
\(L(1)\) \(\approx\) \(0.8615431271 + 0.07142336024i\)
\(L(1)\) \(\approx\) \(0.8615431271 + 0.07142336024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−32.32122488271559241175413688024, −31.13615424100686174506551447538, −29.75492422760048571427717966189, −28.93296387063081063004390541344, −27.34194552840490146191943076635, −26.34397087702428861375534466041, −25.50181715303365113399553453765, −24.39094605331804964737406262899, −23.61189173952552000061638028280, −22.48248423516712644496143776719, −20.429456317864161682630273493915, −19.33380059720804007928000975869, −18.60001004866666635876061559180, −17.20200269255370000659513240923, −16.35339752359155581511534278521, −14.65329108398390780721306226677, −13.88801967373127014738295979548, −12.67843535907272444793774636930, −10.717135568236501818882424948809, −9.24567377828367583036663847316, −8.07731086762784508591832862088, −6.977087557221896131402583925960, −5.89999741126418323913572070000, −3.64563703007843048773639466111, −1.210959795943881323931123406180, 2.32916536639200035698130087984, 3.57974579525258156591908144132, 5.10242005017280119988957607028, 7.52092548514733843129534188107, 9.218958991203555385808000578483, 9.51869533284427186397475579409, 11.09165021212674927744468449862, 12.27140601524550044445399492009, 13.68494927072188841445705365709, 15.17486855215578344134637551703, 16.32654415060813490864372342196, 17.65934902009074894839215348339, 19.01024462837097913332402781084, 19.9537484764539592303203859057, 20.95047284321697220644949707841, 22.005693239134584814645050574365, 22.76678966384344708026310499989, 25.11599739901318771044225735625, 25.79710475586722500485421136026, 26.99427117655407755663138195946, 27.938040484989343999651129768016, 28.67050653898517170160896826381, 30.1176969499113893282653362193, 31.25409942088549943287289144686, 31.931838172887132543553732553464

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