Properties

Label 1-161-161.135-r1-0-0
Degree $1$
Conductor $161$
Sign $0.515 + 0.856i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (0.723 + 0.690i)2-s + (0.981 − 0.189i)3-s + (0.0475 + 0.998i)4-s + (0.786 − 0.618i)5-s + (0.841 + 0.540i)6-s + (−0.654 + 0.755i)8-s + (0.928 − 0.371i)9-s + (0.995 + 0.0950i)10-s + (−0.723 + 0.690i)11-s + (0.235 + 0.971i)12-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (−0.995 + 0.0950i)16-s + (0.888 − 0.458i)17-s + (0.928 + 0.371i)18-s + (0.888 + 0.458i)19-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (0.981 − 0.189i)3-s + (0.0475 + 0.998i)4-s + (0.786 − 0.618i)5-s + (0.841 + 0.540i)6-s + (−0.654 + 0.755i)8-s + (0.928 − 0.371i)9-s + (0.995 + 0.0950i)10-s + (−0.723 + 0.690i)11-s + (0.235 + 0.971i)12-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (−0.995 + 0.0950i)16-s + (0.888 − 0.458i)17-s + (0.928 + 0.371i)18-s + (0.888 + 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.515 + 0.856i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.515 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.758582768 + 2.124852693i\)
\(L(\frac12)\) \(\approx\) \(3.758582768 + 2.124852693i\)
\(L(1)\) \(\approx\) \(2.256336439 + 0.8511615084i\)
\(L(1)\) \(\approx\) \(2.256336439 + 0.8511615084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.723 + 0.690i)T \)
3 \( 1 + (0.981 - 0.189i)T \)
5 \( 1 + (0.786 - 0.618i)T \)
11 \( 1 + (-0.723 + 0.690i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.888 - 0.458i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.928 + 0.371i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.580 - 0.814i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (-0.981 - 0.189i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.0475 + 0.998i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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

−27.370004219117693713473269419, −26.38330683983681618479532335311, −25.38442657818453512353587447382, −24.545932011319223364546298094610, −23.35698461430319651550923377876, −22.23111799932735875799696930181, −21.35883075134845071916897262082, −20.79935279931712175706943461601, −19.67676609940421922961459403639, −18.75662598168253349459171503479, −17.923531647816213053204022527300, −15.97849531231395241750869490275, −15.101993560107707776595657569457, −14.03181692256584797682270352660, −13.54530601388014692324425729314, −12.48021887784427701454811711388, −10.79099534173050997178789051990, −10.23428512766681920348993060004, −9.11530116845647364623975727653, −7.682047834259393109407605167382, −6.13092394577999714345858395555, −5.05624779235686910854232245542, −3.33623678269059051197247074725, −2.83549011768772866489365505493, −1.36114731044111858227195804674, 1.73154208191277791908515870835, 3.03863042220320381235890252064, 4.41683932649938175173764843417, 5.521171095750510202586157970063, 6.89119724549922409717058336007, 7.90893572506435637347749358975, 8.97425364376534707484869797061, 9.958297790434010827754883366927, 12.012080758905266124599892768551, 12.8972626571041391048061957670, 13.81419161570735900798426507509, 14.40279719160293234637458481215, 15.6964891993566951883313353724, 16.47785732998702124067929111112, 17.76509521053825025884271117713, 18.67805017711173883544492455207, 20.37238721090217471762303616069, 20.86084328755562343941030678118, 21.67957117435744500739092557873, 23.05323439701817658173732825990, 24.06963072075980373132684298219, 24.7851768960767298378334800477, 25.75759452391976717156984196748, 26.105638035348520334316926187440, 27.477792231652050917868340432293

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