Properties

Label 1-65-65.3-r1-0-0
Degree $1$
Conductor $65$
Sign $-0.690 + 0.723i$
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  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s i·12-s − 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + i·18-s + (0.5 + 0.866i)19-s + 21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s i·12-s − 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + i·18-s + (0.5 + 0.866i)19-s + 21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.690 + 0.723i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ -0.690 + 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5125940037 + 1.197411113i\)
\(L(\frac12)\) \(\approx\) \(0.5125940037 + 1.197411113i\)
\(L(1)\) \(\approx\) \(0.9672035042 + 0.5284296765i\)
\(L(1)\) \(\approx\) \(0.9672035042 + 0.5284296765i\)

Euler product

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

−31.761962941642766644353404116707, −30.23589740560603565069912317327, −29.16167414530608579029402995345, −28.70138421451477969421829907230, −27.312885182821131082733091449578, −26.11995150141457194065695508414, −24.3258201901021529221808856218, −23.47319027671087238623140951724, −22.4167550775094448675875818936, −21.7270400823043025013001667504, −20.48018022260096852075107419787, −19.333668023853324683813860656586, −17.87138629115403741686594601032, −16.19308186657015224925262964571, −15.67664769922084317592473860907, −13.89947603977856469114880928287, −12.870216786380335362194749782898, −11.54514672113834302323757498225, −10.61340197350599096022930282083, −9.48462288321028369132247231242, −6.87686194760521846777095654437, −5.76180648528618091931506071089, −4.4381278399772691596784137858, −3.07897389785157763310440644611, −0.55131570709625579190672329227, 2.37669733869627709197959212004, 4.38286064490164692859649165963, 5.80083277964727877438986423304, 6.68792939096067492965966139304, 8.04066587640755825574771008183, 10.16878857135937968702604511410, 11.812712147591674455953772185374, 12.608324538313098696729479980086, 13.58688598768088403777184008336, 15.29512661603756972716364939087, 16.185914863829118839444125129808, 17.36151821845176085914648038885, 18.50884120009137576675717316428, 20.07085097704010886747871817079, 21.60619197123922407907218078592, 22.56439825524119507631736664622, 23.24216553749734142307717447127, 24.4516858792258535267399649768, 25.32810327520158647846128483327, 26.53468694233219331768881067974, 28.36846031415455206013001769571, 29.04830207483297729126358287012, 30.29559409585395073876334685429, 31.18800384206185958771772651103, 32.4040394694906054545072453347

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