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Properties

Label 1-775-775.163-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.920 + 0.391i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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  + (−0.951 − 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.951 − 0.309i)7-s + (−0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.951 + 0.309i)13-s − 14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + (−0.587 − 0.809i)18-s + (−0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.951 − 0.309i)7-s + (−0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 + 0.809i)12-s + (−0.951 + 0.309i)13-s − 14-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + (−0.587 − 0.809i)18-s + (−0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.920 + 0.391i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.920 + 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1333273773 + 0.6542167094i\)
\(L(\frac12)\) \(\approx\) \(0.1333273773 + 0.6542167094i\)
\(L(1)\) \(\approx\) \(0.8557404125 + 0.1133028792i\)
\(L(1)\) \(\approx\) \(0.8557404125 + 0.1133028792i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + iT \)
29 \( 1 - T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.951 + 0.309i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−21.31838078545059417082381018258, −20.79054918132444639539672707209, −20.215342989912578283670350074240, −19.094851128733786523407423161536, −18.64518251898141201589813315991, −18.01530054146254643752005757695, −17.03794796842101414041640388635, −16.141112639943084057293908813779, −15.24249685906562151626110437456, −14.56179356074591357824417794032, −14.00749848864104312038915823565, −12.67455037950831399389015324154, −11.88494527295591082042755297804, −10.785860742391822077287915208990, −9.98963434434379254054288823907, −9.12431284445337712287408918492, −8.150013336099734021012956634722, −7.87447232134617045326257874121, −6.94680996747218336672181937733, −5.72130380030717963401216141756, −4.81222342367694294882337503888, −3.176951651987220123368313351097, −2.33941655455974471010046098547, −1.458819618705965389783494935478, −0.16079243248200308980422270336, 1.55349556686568168143517449549, 2.16169207810719887130794390140, 3.229125442545808376048033150011, 4.29582874691580868320998982664, 5.29882925893755434050327153639, 7.06346903507813453810261988401, 7.57752232716224172457563917622, 8.303256002635046238677673197664, 9.18826693386401584758439783887, 10.00937043715326663766921275834, 10.64463312527370522300818311748, 11.58855361397274169588448731849, 12.63354982111134586321959102274, 13.45468216006479842135530974224, 14.66039032561078528276239802697, 15.15079903936555082382314261443, 16.007245912490730326538099580858, 17.128860872010122012487485588906, 17.601798061351991683090935920780, 18.66051786726221712015216073589, 19.33149843289176357735583806645, 20.0826029804331036983582495368, 20.75184988792960028638634846623, 21.39280264803754980724553858948, 22.01611268361214169783181153487

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