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Properties

Label 1-69-69.38-r0-0-0
Degree $1$
Conductor $69$
Sign $0.764 + 0.644i$
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  + (0.959 + 0.281i)2-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.415 − 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.415 + 0.909i)10-s + (−0.959 + 0.281i)11-s + (0.415 − 0.909i)13-s + (−0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.654 + 0.755i)20-s − 22-s + (−0.959 − 0.281i)25-s + (0.654 − 0.755i)26-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.415 − 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.415 + 0.909i)10-s + (−0.959 + 0.281i)11-s + (0.415 − 0.909i)13-s + (−0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (−0.654 + 0.755i)20-s − 22-s + (−0.959 − 0.281i)25-s + (0.654 − 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(0.320434\)
Root analytic conductor: \(0.320434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (0:\ ),\ 0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.377938426 + 0.5032153484i\)
\(L(\frac12)\) \(\approx\) \(1.377938426 + 0.5032153484i\)
\(L(1)\) \(\approx\) \(1.497923106 + 0.3812671157i\)
\(L(1)\) \(\approx\) \(1.497923106 + 0.3812671157i\)

Euler product

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

−31.73391869666421496660240286791, −31.01278158255213694336585770935, −29.50661327860750454316731493317, −28.57243161795002295421744670136, −27.91962018142597418651897304930, −25.93791506699985795316959960291, −24.90209294308098071488996444712, −23.86788724597431353381957395310, −23.0528459169545313067407587704, −21.40609450997760720341155527507, −21.09950690773700674539283330027, −19.59203223771486213038273464965, −18.64955539727148244852119121247, −16.57158297290392375106986898013, −15.84034677666190876619865632287, −14.56358905329548413198528022595, −13.06437915589755892269146201137, −12.44748637509047669972940995232, −11.18207390114359311505688491320, −9.59066027977716646664127399093, −8.11143060697423245535834361267, −6.16601445759224821934210875586, −5.15760963915594365849335081010, −3.68429268297079273548229110118, −1.95354316575904223609827617475, 2.71256253902432592314717151711, 3.85535792105064962780380100994, 5.54435445472554231933230711262, 6.94654265525579188393662983138, 7.80547780273660577088821574584, 10.26655540541446648592102953668, 11.10284672644128410251800238965, 12.74437814802936570016273337101, 13.678640545564170070478381393188, 14.88116869887037289439033668173, 15.81597843994239708088872520234, 17.1210036629957997336706135248, 18.538916294070923517380226614097, 20.00469212885236328962554433005, 20.9884228055221157183924248658, 22.37116540974269782030749939326, 23.086939674898868441637212100091, 23.905098924510290528678325377300, 25.688036726869434397901435797633, 25.995485362609902668786867452330, 27.48187362104359267544923620962, 29.27293076044455545941056800932, 29.95002660786973928454729967565, 30.86690972492260914619378385967, 32.01812046447165901881754587151

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