↑

Properties

Label 1-167-167.9-r0-0-0
Degree $1$
Conductor $167$
Sign $0.961 + 0.274i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.316 + 0.948i)2-s + (−0.965 + 0.261i)3-s + (−0.800 − 0.599i)4-s + (0.672 + 0.739i)5-s + (0.0567 − 0.998i)6-s + (−0.644 − 0.764i)7-s + (0.822 − 0.569i)8-s + (0.862 − 0.505i)9-s + (−0.914 + 0.404i)10-s + (−0.243 − 0.969i)11-s + (0.929 + 0.369i)12-s + (−0.584 − 0.811i)13-s + (0.929 − 0.369i)14-s + (−0.843 − 0.537i)15-s + (0.280 + 0.959i)16-s + (0.988 + 0.150i)17-s + ⋯
L(s)  = 1  + (−0.316 + 0.948i)2-s + (−0.965 + 0.261i)3-s + (−0.800 − 0.599i)4-s + (0.672 + 0.739i)5-s + (0.0567 − 0.998i)6-s + (−0.644 − 0.764i)7-s + (0.822 − 0.569i)8-s + (0.862 − 0.505i)9-s + (−0.914 + 0.404i)10-s + (−0.243 − 0.969i)11-s + (0.929 + 0.369i)12-s + (−0.584 − 0.811i)13-s + (0.929 − 0.369i)14-s + (−0.843 − 0.537i)15-s + (0.280 + 0.959i)16-s + (0.988 + 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.961 + 0.274i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.961 + 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6171163088 + 0.08634444163i\)
\(L(\frac12)\) \(\approx\) \(0.6171163088 + 0.08634444163i\)
\(L(1)\) \(\approx\) \(0.6322454534 + 0.2023150106i\)
\(L(1)\) \(\approx\) \(0.6322454534 + 0.2023150106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.316 + 0.948i)T \)
3 \( 1 + (-0.965 + 0.261i)T \)
5 \( 1 + (0.672 + 0.739i)T \)
7 \( 1 + (-0.644 - 0.764i)T \)
11 \( 1 + (-0.243 - 0.969i)T \)
13 \( 1 + (-0.584 - 0.811i)T \)
17 \( 1 + (0.988 + 0.150i)T \)
19 \( 1 + (-0.387 - 0.922i)T \)
23 \( 1 + (0.726 + 0.686i)T \)
29 \( 1 + (0.726 - 0.686i)T \)
31 \( 1 + (0.898 - 0.438i)T \)
37 \( 1 + (0.862 + 0.505i)T \)
41 \( 1 + (0.614 - 0.788i)T \)
43 \( 1 + (-0.982 - 0.188i)T \)
47 \( 1 + (-0.999 + 0.0378i)T \)
53 \( 1 + (-0.942 + 0.334i)T \)
59 \( 1 + (0.988 - 0.150i)T \)
61 \( 1 + (-0.0189 - 0.999i)T \)
67 \( 1 + (0.672 - 0.739i)T \)
71 \( 1 + (0.974 - 0.225i)T \)
73 \( 1 + (0.280 - 0.959i)T \)
79 \( 1 + (-0.752 - 0.658i)T \)
83 \( 1 + (-0.316 - 0.948i)T \)
89 \( 1 + (-0.843 + 0.537i)T \)
97 \( 1 + (0.898 + 0.438i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.06883528280232256425009320201, −26.99560550021616234572040891812, −25.5819490841715532044419271897, −24.85969110906573740604750630774, −23.38298661974850052371656858330, −22.65725373926677934307693339831, −21.533375246251830306049475785222, −21.06918102940634582394775707349, −19.66182854090314197789657036194, −18.68206760663666820856076653986, −17.912512192073514471342201836277, −16.877222293609065414528881656954, −16.26282527123544114225569014968, −14.39296470021302084528050289637, −12.834143836607694619373220804824, −12.55058243054448743879283034554, −11.65493043833624678536715821269, −10.0871237068148658198451045451, −9.67405171739139453663681783182, −8.29970390934153688397695527709, −6.73173111055529018148299992025, −5.360780159413598561072610645474, −4.49001221966077152590505319699, −2.51180190028945266011971109384, −1.34310608442402976520903790064, 0.74150904521224941587072577736, 3.30744345688452233631374655514, 4.92859383818497320246727821168, 5.97393661808458465354095715778, 6.70150530048383764824455555376, 7.80062251454860171060816310522, 9.590832527942813278345497922195, 10.20968731574541232198588121859, 11.113553573231257824407825580842, 12.9430771644384055411630122291, 13.74533856209125147670456910965, 15.00926844861471241217681954688, 15.92537796316279205332539819581, 17.068300391095244242858788131921, 17.397721906752139910585882399096, 18.61342038081171287069045693037, 19.423402595631553403486476294802, 21.27513189260686714270163599752, 22.16198912312700058924650847857, 22.975935878416538336763175276623, 23.66746908524325461660081169151, 24.83861706589560277897479404993, 25.8825271834404606895938563255, 26.713115660555950872233418739032, 27.36577856215850051511663486350

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