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Properties

Label 1-775-775.184-r1-0-0
Degree $1$
Conductor $775$
Sign $0.461 - 0.887i$
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  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + 12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + (−0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)6-s + (0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + 12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + (−0.309 + 0.951i)18-s + (0.309 − 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.461 - 0.887i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ 0.461 - 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.666579499 - 2.225504984i\)
\(L(\frac12)\) \(\approx\) \(3.666579499 - 2.225504984i\)
\(L(1)\) \(\approx\) \(1.979839579 - 0.5079140339i\)
\(L(1)\) \(\approx\) \(1.979839579 - 0.5079140339i\)

Euler product

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

−22.46527892483529636003052366729, −21.51438791150969762346102685374, −20.90198573639130393603030476270, −19.921207209326937181918184239405, −19.03080668759165185350269549855, −18.205543624743814379137863334714, −17.3412577681433107698861447904, −16.716983881981159390097961839353, −15.60825521761343322851530189549, −14.65085411677100755106797333578, −14.142916472732446169203816076506, −13.5922685668701501704026060762, −12.348625254912037581282329462540, −11.8698025870593829784455003581, −11.32707432929930696570769742798, −9.46832598259143963364003781150, −8.63161948072199537022707959772, −7.78260605539100061281921690176, −7.14176549718541289535975674120, −6.1271667690619518128413991400, −5.426479401168713039313259410674, −4.33905903198296431570442668555, −3.158474040694879321658350783542, −2.30007855363597358099631410165, −1.1803488876103896098652444339, 0.730474296826028565767117820461, 2.01952207345337666567378358685, 2.99611623421405116408034399229, 4.05542674390351756321969960933, 4.63890905029423699788378749960, 5.37475584273811945443691782604, 6.62431349424620458911442103067, 7.709508028344423799446401195071, 8.88465390836994057213043780714, 9.79453411527446437082985948657, 10.547160052939875522385602637016, 11.14347332030396644167606644610, 12.12176964336282010456945905864, 12.98128584638385320212075174789, 14.11994494294003391319344192766, 14.62671978052299519242134181969, 15.10562163045136333545307868914, 16.16347148435960271944253001806, 17.13430505090645791983425443539, 17.86770638084863594624746511739, 19.3097273703455444413159326476, 20.011560054882449793386169194345, 20.30281546123454413354459118005, 21.47423769746186784993144676676, 21.70342160975455477744941463402

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