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Properties

Label 1-77-77.24-r0-0-0
Degree $1$
Conductor $77$
Sign $0.808 + 0.587i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.808 + 0.587i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.808 + 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8406974657 + 0.2731800534i\)
\(L(\frac12)\) \(\approx\) \(0.8406974657 + 0.2731800534i\)
\(L(1)\) \(\approx\) \(0.9196500077 + 0.2031795045i\)
\(L(1)\) \(\approx\) \(0.9196500077 + 0.2031795045i\)

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

−31.14610950933427270280139428596, −29.99556186087838851066682182567, −28.88054488668012406940976609311, −27.84994463704120146471713226799, −26.90629498121278440740632075505, −25.93546602287636432280279888849, −24.90847211099217812601186257754, −24.14049207324646902297422748052, −21.90866372596963904032967711848, −20.97184042008757408406799810869, −20.11846319611519512013006710524, −19.32840881927045792368139483676, −18.07460673314366034789954296998, −16.66251204640140288025738924480, −15.91869609537698042275553811226, −14.34574584973031087057403572868, −12.93197091301434679647108243664, −11.8818612460793777106010364304, −10.11029646380673856669776455690, −9.25519487906222818225135745019, −8.31248289487468820425943493927, −7.12805130420861605672335124611, −4.74424697697348573701475329016, −3.09408260600771876590168807003, −1.57992341387840411782284677573, 1.94405889788413251251938742741, 3.31978586773099274497471833575, 5.8467012543475967381524582181, 7.33056613064629581835877621719, 7.964396457364982736691726541263, 9.602622245602518176958761337050, 10.29044780998396339784303488360, 11.958805292368458824441793809493, 13.82990819795933214564538990761, 14.77724507698132911452154167000, 15.56957565473373073805526002553, 17.18639954902215638532803296005, 18.36149227162872630782726914371, 19.10601188693525077477025064740, 20.07112980659551774255286739013, 21.30038121099317273673238686765, 22.86913129359824596942470538450, 24.25372643721155528372721149299, 25.1982922054422131636767483274, 26.05755090325791665068397747848, 26.81918689190506805679285756565, 27.7898568519067534282835368003, 29.519456756138877336294925539077, 29.9902286000681655484547324801, 31.4036645624177693313923299585

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