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Properties

Label 1-167-167.4-r0-0-0
Degree $1$
Conductor $167$
Sign $0.750 - 0.661i$
Analytic cond. $0.775544$
Root an. cond. $0.775544$
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  + (−0.169 + 0.985i)2-s + (−0.316 + 0.948i)3-s + (−0.942 − 0.334i)4-s + (−0.993 + 0.113i)5-s + (−0.881 − 0.472i)6-s + (0.672 − 0.739i)7-s + (0.489 − 0.872i)8-s + (−0.800 − 0.599i)9-s + (0.0567 − 0.998i)10-s + (−0.999 + 0.0378i)11-s + (0.614 − 0.788i)12-s + (−0.644 − 0.764i)13-s + (0.614 + 0.788i)14-s + (0.206 − 0.978i)15-s + (0.776 + 0.629i)16-s + (−0.965 − 0.261i)17-s + ⋯
L(s)  = 1  + (−0.169 + 0.985i)2-s + (−0.316 + 0.948i)3-s + (−0.942 − 0.334i)4-s + (−0.993 + 0.113i)5-s + (−0.881 − 0.472i)6-s + (0.672 − 0.739i)7-s + (0.489 − 0.872i)8-s + (−0.800 − 0.599i)9-s + (0.0567 − 0.998i)10-s + (−0.999 + 0.0378i)11-s + (0.614 − 0.788i)12-s + (−0.644 − 0.764i)13-s + (0.614 + 0.788i)14-s + (0.206 − 0.978i)15-s + (0.776 + 0.629i)16-s + (−0.965 − 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.750 - 0.661i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.750 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2475724293 - 0.09350634869i\)
\(L(\frac12)\) \(\approx\) \(0.2475724293 - 0.09350634869i\)
\(L(1)\) \(\approx\) \(0.4692082786 + 0.2350623338i\)
\(L(1)\) \(\approx\) \(0.4692082786 + 0.2350623338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.169 + 0.985i)T \)
3 \( 1 + (-0.316 + 0.948i)T \)
5 \( 1 + (-0.993 + 0.113i)T \)
7 \( 1 + (0.672 - 0.739i)T \)
11 \( 1 + (-0.999 + 0.0378i)T \)
13 \( 1 + (-0.644 - 0.764i)T \)
17 \( 1 + (-0.965 - 0.261i)T \)
19 \( 1 + (0.954 - 0.298i)T \)
23 \( 1 + (-0.243 - 0.969i)T \)
29 \( 1 + (-0.243 + 0.969i)T \)
31 \( 1 + (-0.700 + 0.713i)T \)
37 \( 1 + (-0.800 + 0.599i)T \)
41 \( 1 + (-0.0189 - 0.999i)T \)
43 \( 1 + (0.898 - 0.438i)T \)
47 \( 1 + (-0.752 - 0.658i)T \)
53 \( 1 + (-0.982 - 0.188i)T \)
59 \( 1 + (-0.965 + 0.261i)T \)
61 \( 1 + (0.351 - 0.936i)T \)
67 \( 1 + (-0.993 - 0.113i)T \)
71 \( 1 + (-0.387 - 0.922i)T \)
73 \( 1 + (0.776 - 0.629i)T \)
79 \( 1 + (-0.455 + 0.890i)T \)
83 \( 1 + (-0.169 - 0.985i)T \)
89 \( 1 + (0.206 + 0.978i)T \)
97 \( 1 + (-0.700 - 0.713i)T \)
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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

−28.08969380710628534590017221343, −26.997958028453228343240741918805, −26.085468306751529325492328752884, −24.462703011649546412284272245508, −23.89855425017372788469645618388, −22.86871976209843727177517646280, −21.92918889395455152711454862928, −20.78759062353374804342771935882, −19.72923867566739746993401090932, −18.967933918587223829387840381357, −18.209028408554761276368511229386, −17.350279755216429202790053011497, −15.95411058528631339703864964917, −14.57362271468261780262435694898, −13.37018030065624474318192787550, −12.40615825846962333190686241910, −11.59589893337860911682258223173, −11.05067153505275286196691012521, −9.31623907787086850928729531581, −8.11422219270452485470132446350, −7.50017745163202197188230407731, −5.552484949881252217913381252, −4.45872518225275273941103844804, −2.78569614674192692820732697103, −1.67995509644276784912959876105, 0.241016074793398066809148796122, 3.36983513487834145857785907953, 4.66183795230752027786339663951, 5.19340573049148275026212177570, 6.9568347884727121274889977517, 7.86181914663772806867880650198, 8.88280745335079501579046050709, 10.32673484962703823957507175122, 10.94809524480344782569691806691, 12.459782227387731709232343501349, 13.955060621242614836553021599100, 14.93529821677197813913542131396, 15.69230924690040639159492495710, 16.40487677244288447615108988359, 17.53562568383085419478071952895, 18.284224581956443512846847427456, 19.86682333647869629713787420578, 20.57819820546519571268311512213, 22.139440599518665287329971454544, 22.75124553641941141441712187972, 23.7602170488014344106857275115, 24.3553412384737672795047044311, 25.957252805985385410932486565563, 26.73630501227460989152168670206, 27.17625824950508228512548002615

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