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Properties

Label 1-69-69.14-r0-0-0
Degree $1$
Conductor $69$
Sign $0.0174 + 0.999i$
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  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (0.654 − 0.755i)10-s + (0.841 + 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.142 + 0.989i)20-s − 22-s + (0.841 − 0.540i)25-s + (0.142 − 0.989i)26-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (0.654 − 0.755i)10-s + (0.841 + 0.540i)11-s + (−0.654 + 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (−0.415 + 0.909i)19-s + (−0.142 + 0.989i)20-s − 22-s + (0.841 − 0.540i)25-s + (0.142 − 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.0174 + 0.999i$
Analytic conductor: \(0.320434\)
Root analytic conductor: \(0.320434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (0:\ ),\ 0.0174 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4025361303 + 0.3955869120i\)
\(L(\frac12)\) \(\approx\) \(0.4025361303 + 0.3955869120i\)
\(L(1)\) \(\approx\) \(0.5949408789 + 0.2848442032i\)
\(L(1)\) \(\approx\) \(0.5949408789 + 0.2848442032i\)

Euler product

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

−31.3430845113279519303180029222, −30.208241459046460146719113953898, −29.56092660430121307856580182689, −27.98430271987152079814774581171, −27.303043872864401340587312627477, −26.63130559142641400312276172636, −25.06655135307265171234600307696, −24.047334194174373917445420764512, −22.658254742334272563299960294032, −21.31199272048150534504053371367, −20.03047585727433679253436896171, −19.61144919105697733286695821933, −18.135885167084961781797801264763, −17.01711152352820173562617645204, −16.08484076873491499065353741260, −14.55911894465751850506128753576, −12.84483996906254130795022635093, −11.61839443657721991344326816885, −10.81919639883970437045592058789, −9.250872197448729643811651275840, −8.02116693167184603254743865921, −7.09213010458559384774309357540, −4.58448608098081294817842887779, −3.17563253665816250225279113004, −0.96232529288932026425437074967, 1.956854261784113560006561567285, 4.32241025153315076046534592527, 6.06698271625630670537347745582, 7.45485707256871330040134563943, 8.44037817829674051986894639637, 9.741135844622269325723308305130, 11.270206793107503950819598859753, 12.14607366258038728997767902013, 14.64395095540596760351072159392, 14.95041345179794517896176548209, 16.41461823817766312923896813622, 17.458426386629060991123161798753, 18.80957058643508463842610847725, 19.405072946220673782092382510867, 20.780721871934710206591127482906, 22.37006472174596003399796238367, 23.65342752155071504312291473991, 24.50891740711789645953770663627, 25.61891761786768845703201945154, 26.82930028618820665035514230325, 27.65048919783941068303471336221, 28.39297722297770265929463489229, 29.92172134467860378015308594987, 31.10041439932825405211259233870, 32.172162349357803818836520340809

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