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Properties

Label 1-69-69.2-r1-0-0
Degree $1$
Conductor $69$
Sign $0.378 + 0.925i$
Analytic cond. $7.41507$
Root an. cond. $7.41507$
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  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + (−0.654 + 0.755i)19-s + (0.959 − 0.281i)20-s + 22-s + (0.415 + 0.909i)25-s + (0.959 − 0.281i)26-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (−0.142 + 0.989i)10-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + (−0.654 + 0.755i)19-s + (0.959 − 0.281i)20-s + 22-s + (0.415 + 0.909i)25-s + (0.959 − 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.378 + 0.925i$
Analytic conductor: \(7.41507\)
Root analytic conductor: \(7.41507\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 69,\ (1:\ ),\ 0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3045172176 + 0.2045272664i\)
\(L(\frac12)\) \(\approx\) \(0.3045172176 + 0.2045272664i\)
\(L(1)\) \(\approx\) \(0.5464928499 - 0.1738640890i\)
\(L(1)\) \(\approx\) \(0.5464928499 - 0.1738640890i\)

Euler product

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

−31.802569788669165042740451938565, −30.47782183154511796218874282370, −28.9665340610073924987104716209, −27.693461091087701639902313641378, −27.07483313184785701723578051468, −25.85880446470623031440137295115, −24.90718513739537517694471011675, −23.7666653118581111686455438420, −22.79369239719686826452000080137, −21.71601441798008427200318550198, −19.785640867634808473162366602478, −18.785000472068479918710651767795, −18.050118707811245758762878446624, −16.43100138893861229251062512207, −15.49847847540921829593689479496, −14.723491274945997063596600032046, −13.18680701721911723398250867270, −11.564063519780019386559445653468, −10.21537926475686330760926569504, −8.67604187554937166889866630620, −7.75498121407437010944945089673, −6.3298243903386481267019363925, −5.056692032962853973567038238963, −3.046254424380743597183650843539, −0.22272967373836176981798339233, 1.56636609866453987841111443876, 3.67657999812304108789888703834, 4.62542020349987665815092429613, 7.23094474738210972482947068487, 8.30246201793445477897299218667, 9.7458547899604849891909930275, 10.85719252664905441240859192941, 12.165093631623655888542730315157, 13.00739863294234763932311057408, 14.5428302702462413862018051204, 16.3618273451180834328118621106, 17.110933267882072931612320822557, 18.61590368935399954731173534340, 19.678817049620107408537956995732, 20.42864103007609278700660870349, 21.49983655275671384270895861906, 23.11192626242612301865944032573, 23.649935592041386878012920662124, 25.58548888501943513456978573744, 26.60719191309940784894538391921, 27.56458291245267139937318214134, 28.49198212921696127324084956286, 29.52578841590835137619285651632, 30.702985412136030318968070900325, 31.45182203697803533078267991602

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