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Properties

Label 1-65-65.34-r1-0-0
Degree $1$
Conductor $65$
Sign $0.957 - 0.289i$
Analytic cond. $6.98522$
Root an. cond. $6.98522$
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  − i·2-s − 3-s − 4-s + i·6-s + i·7-s + i·8-s + 9-s − i·11-s + 12-s + 14-s + 16-s + 17-s − i·18-s + i·19-s − i·21-s − 22-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s + i·6-s + i·7-s + i·8-s + 9-s − i·11-s + 12-s + 14-s + 16-s + 17-s − i·18-s + i·19-s − i·21-s − 22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ 0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9618253774 - 0.1424162708i\)
\(L(\frac12)\) \(\approx\) \(0.9618253774 - 0.1424162708i\)
\(L(1)\) \(\approx\) \(0.7458936127 - 0.2258384142i\)
\(L(1)\) \(\approx\) \(0.7458936127 - 0.2258384142i\)

Euler product

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

−32.51317334665635314172374964486, −30.87825942997694310324093227558, −29.81075613418182490242524038849, −28.39460883706144653953883344497, −27.48896430747447162538204760469, −26.45677220221986172448187393187, −25.27395755241788463771622988919, −23.936160431749305427787788476903, −23.21949043563548370495902390959, −22.42096847618908576340868266163, −20.99552329301388123049556712606, −19.22422847063423071241369132518, −17.80672974148313918871497411330, −17.16169288099476577852960515558, −16.141994341756104032531097005068, −14.934078435743885548634592579704, −13.4980613363765326109801648098, −12.36903621334040375840478603012, −10.65777440502465297241093344496, −9.529560384116313765981244367611, −7.552833981021442199681789348521, −6.75530344114179054192415990032, −5.24969334407576720286573414775, −4.15575356800328867305461726051, −0.740747383397072730069639969834, 1.191641341121034153441063143485, 3.16863520459971308663654219633, 4.97155439431867733914609638305, 6.04230438552323036778305369652, 8.274157431179052087156412198520, 9.71953464528201339863751568466, 10.939644476572307599441884444680, 11.94586847977110777291177707501, 12.79153781213157393941153276636, 14.359358825352863186858369384112, 16.04741008650909299510279212422, 17.29082425065281219370401640766, 18.54644329046427864991479182734, 19.10961837434595071302800628434, 21.01657466241597828481263744844, 21.66814110548880393249378660998, 22.71531226587613778221840444779, 23.71493550859033073296405435140, 25.124692547379610711377595732089, 26.9729886824787562045106771697, 27.632618364042944112862586329078, 28.789241959981170194513583034047, 29.36591786598184437854305385788, 30.56203561519578279507008524426, 31.74385397317974639682551300991

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