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Properties

Label 1-69-69.56-r0-0-0
Degree $1$
Conductor $69$
Sign $-0.898 - 0.438i$
Analytic cond. $0.320434$
Root an. cond. $0.320434$
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  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.841 + 0.540i)10-s + (−0.142 − 0.989i)11-s + (0.841 − 0.540i)13-s + (−0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)17-s + (0.959 + 0.281i)19-s + (0.415 + 0.909i)20-s − 22-s + (−0.142 + 0.989i)25-s + (−0.415 − 0.909i)26-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.841 + 0.540i)10-s + (−0.142 − 0.989i)11-s + (0.841 − 0.540i)13-s + (−0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)17-s + (0.959 + 0.281i)19-s + (0.415 + 0.909i)20-s − 22-s + (−0.142 + 0.989i)25-s + (−0.415 − 0.909i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.898 - 0.438i$
Analytic conductor: \(0.320434\)
Root analytic conductor: \(0.320434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (0:\ ),\ -0.898 - 0.438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1541437805 - 0.6682594926i\)
\(L(\frac12)\) \(\approx\) \(0.1541437805 - 0.6682594926i\)
\(L(1)\) \(\approx\) \(0.5616108486 - 0.5906178190i\)
\(L(1)\) \(\approx\) \(0.5616108486 - 0.5906178190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (-0.654 - 0.755i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
11 \( 1 + (-0.142 - 0.989i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (0.654 - 0.755i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 - T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (-0.415 + 0.909i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (-0.654 + 0.755i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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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

−32.48482813605904726668533390067, −31.00114154230996738437799150670, −30.88815158502529390550716063283, −28.88551360548211736564592313559, −27.76820499125254027740692567878, −26.491828721322873948441440598274, −25.85944637029561373391270492540, −24.74081541325799120390902189969, −23.34683412692181595904925920660, −22.714858945172741140495057254025, −21.71008433707437108648901926235, −19.84382633656310730896684753233, −18.59564834024385696378821942467, −17.80587521380056197993739300071, −16.01731066877190950991453704882, −15.58386729190441112986768419897, −14.311131135750909671951765410457, −13.03337264816695712332304320672, −11.71693242605339136456495789494, −9.92470199341920128718324695833, −8.65159084137207308572049568089, −7.1641901260026163170881684606, −6.34716197607530297087446811779, −4.58691885862096126379365235705, −3.11035396470579966014705791309, 0.832490785661800432812968484277, 3.17078768924538995768467426865, 4.258134246187416620108165839, 5.9015270493637994151136086531, 8.06375545511597503785250837975, 9.1987644912768807500330916066, 10.60148145081496106013694000403, 11.69628707136188476918159355334, 13.00187546611630162636289980018, 13.67503723506281522615258089145, 15.57990622197680920646801090382, 16.68202970698281738291913682887, 18.200560187796994062756028479792, 19.46105483629053521822735541642, 20.11228912278943716495881142388, 21.21460311790789676129569156900, 22.56269204628350737351193430565, 23.399677927417753950156143707178, 24.53799728044570462163527375214, 26.342492727370743536148812803885, 27.17346198963185881100897102798, 28.39828299327100784029350641570, 29.10295210329730826214233480778, 30.27825323228528222351544781561, 31.35673047904539831626433151708

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