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Properties

Label 1-65-65.18-r0-0-0
Degree $1$
Conductor $65$
Sign $0.256 + 0.966i$
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  − 2-s + i·3-s + 4-s − i·6-s + 7-s − 8-s − 9-s + i·11-s + i·12-s − 14-s + 16-s + i·17-s + 18-s + i·19-s + i·21-s − i·22-s + ⋯
L(s)  = 1  − 2-s + i·3-s + 4-s − i·6-s + 7-s − 8-s − 9-s + i·11-s + i·12-s − 14-s + 16-s + i·17-s + 18-s + i·19-s + i·21-s − i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(0.301858\)
Root analytic conductor: \(0.301858\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (0:\ ),\ 0.256 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5007106209 + 0.3850950802i\)
\(L(\frac12)\) \(\approx\) \(0.5007106209 + 0.3850950802i\)
\(L(1)\) \(\approx\) \(0.6640832105 + 0.2788922552i\)
\(L(1)\) \(\approx\) \(0.6640832105 + 0.2788922552i\)

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 \( 1 + iT \)
11 \( 1 + T \)
17 \( 1 - iT \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + iT \)
41 \( 1 + iT \)
43 \( 1 \)
47 \( 1 - T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + T \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 + iT \)
83 \( 1 - iT \)
89 \( 1 - iT \)
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

−31.759400371670797167788059875231, −30.44179436937825679538995550200, −29.71828693628592911911572516652, −28.69450856460709711517765745237, −27.546117285616165884659340608290, −26.51769589754611111861941645706, −25.227539824379268763306537338852, −24.39301439530940939436099046787, −23.574421141530289292542170587940, −21.63279645959031702879765902051, −20.353802335422245810570587288588, −19.32290869737854477500205544876, −18.25410997036735979216616513600, −17.54073840974350960378563727845, −16.29044267968787996800131849517, −14.71662538906158652516936461520, −13.40130778856483353647607558575, −11.68957281867834683254489995876, −11.1053147860069023646740601318, −9.144434859242659003394154651307, −8.061151023977091595045472171268, −7.05833111841817324229645553461, −5.571123454991324166945579873497, −2.76013178671656001836929738052, −1.19024827837864578093383102802, 2.083377172677040642588252046808, 4.124288019892223720923175334407, 5.79342123429460273874711398720, 7.67165167473902755047292313780, 8.79213008903843006160667357763, 10.062268463447794471904268025957, 10.94727595991280446345894760445, 12.205330178238784827862086597864, 14.59861819638937969882156363231, 15.264813331690032622674358455792, 16.70394823246946134318187349488, 17.46105696323826205907981828116, 18.74287011815065263454366112002, 20.34578671879663131741981647670, 20.77503404198064944124117459031, 22.10894051241267769939160957856, 23.668920024403687869577075863870, 25.02066129554710157707091319914, 26.068559525935960797395663246953, 27.024642417720435325147919490030, 27.911933406690416219126382056313, 28.59814602509814965501133360762, 30.15239040496786688047065859009, 31.21913530806918119424962596199, 32.83464544947416798699274911046

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