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Properties

Label 1-77-77.72-r1-0-0
Degree $1$
Conductor $77$
Sign $-0.996 - 0.0852i$
Analytic cond. $8.27479$
Root an. cond. $8.27479$
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  + (−0.913 − 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (−0.669 − 0.743i)18-s + (−0.669 + 0.743i)19-s + (−0.809 + 0.587i)20-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (0.809 + 0.587i)6-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.104 + 0.994i)16-s + (−0.913 + 0.406i)17-s + (−0.669 − 0.743i)18-s + (−0.669 + 0.743i)19-s + (−0.809 + 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.996 - 0.0852i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ -0.996 - 0.0852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0009807202416 + 0.02296425311i\)
\(L(\frac12)\) \(\approx\) \(0.0009807202416 + 0.02296425311i\)
\(L(1)\) \(\approx\) \(0.4205456792 + 0.02108629254i\)
\(L(1)\) \(\approx\) \(0.4205456792 + 0.02108629254i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−31.86422686059051380776955577752, −30.05958934399922451199095466180, −28.81893464624652730914187781698, −28.31536684558882930050660751501, −27.43927322391364158908006716262, −26.34945655997137367070978119470, −24.99765443358051026431418589556, −23.965090799628018752704178444978, −23.29901744709104898339139267659, −21.59544659267715894779098638475, −20.5049123870889203718566486760, −19.27520945805674381784399032782, −17.94783710587734845374692438015, −17.155287646467177286243810132246, −16.1253080020456107583454306534, −15.46423989042776572354080634633, −13.44318595595362541352381297719, −11.889821602819398174050109767801, −10.985793243341478477359492370548, −9.57969526142427818531803637100, −8.57103977195215617645860198623, −6.96093165290009109366382987600, −5.76750103409989167845804168422, −4.478530773211989724398019403128, −1.503673322462890603236400463027, 0.01729279640404334955565183068, 1.9582908827186063603557549843, 3.785760293584102417291288464382, 6.08167943121584322557801402960, 7.03718190397061268227556375655, 8.402547768288981915000949751085, 10.278159642678725194881022954478, 10.81656129616581559017492193385, 11.932418863009438979460252893949, 13.17010925896567836795032821755, 15.15772680718228245222866964356, 16.28518809997074784374119413489, 17.48038937464569913115333480414, 18.32501499276363985243146581397, 19.06507522226651111224478898085, 20.52681750197479640553466452063, 21.87503312024791077304276876797, 22.65028053589803488033674689954, 23.98298724890673446851810959983, 25.34224143305639384092023250530, 26.406017241758506266163327477871, 27.44871938565458335247472219211, 28.26502406070505089230408280645, 29.41481698584775443671077004433, 30.10237678704422775357276204175

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