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Properties

Label 1-65-65.48-r1-0-0
Degree $1$
Conductor $65$
Sign $0.955 - 0.293i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.955 - 0.293i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ 0.955 - 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.551259553 - 0.2330116587i\)
\(L(\frac12)\) \(\approx\) \(1.551259553 - 0.2330116587i\)
\(L(1)\) \(\approx\) \(1.101414049 - 0.04009531447i\)
\(L(1)\) \(\approx\) \(1.101414049 - 0.04009531447i\)

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

−31.64554960940056261916524014479, −30.76182724445534462827026964695, −29.87831324345929048348137097020, −28.431667159175101771898223614517, −27.350692876288771577421527960845, −26.708177162136176899345407041871, −25.56521567978437222102155486499, −24.6990278623724316081617306625, −22.90117261415345301163993207164, −21.11532339557252712770233749587, −20.82350653292517624681017295751, −19.67432242333197506117487775989, −18.503291720186764409647969964845, −17.31037127774154040306828698330, −16.04869609557190791825052716326, −14.80915012740775292218984120161, −13.45039575614290713323476016616, −11.880292365452453018864832367086, −10.45163854419782173049003507718, −9.63502682659947017905515903256, −8.164798769605277557717500803065, −7.411358846499888097325632622994, −4.67970656807767272745751884503, −3.14167488719527549263226040976, −1.58828228730185888672784363265, 1.18727734290306079392699140594, 2.78715456592901349322706702621, 5.32184199741746900133833028951, 6.928110549748028053019272037630, 8.20322687507862528928458476785, 8.8445954267281107897934137549, 10.4378334627757450355138171066, 11.90184348076496980626668076571, 13.67958549602665214299372169916, 14.74187654161974963414551601908, 15.715544503873203144701711214108, 17.26672539108407782781264824302, 18.43584447634860528441820994023, 19.090341485940538666135481514689, 20.392757758871029695627409087434, 21.40344014621555755854313195936, 23.59809435522986015224159779310, 24.355400631899760518864776517524, 25.240066625960190358797249319343, 26.32675332753800655881834558194, 27.1826321427485524478374934687, 28.42423871108982111935031872501, 29.61763604881879888337953950303, 30.75413019465097708150139912265, 31.935621178359423357546858338

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