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Properties

Label 1-161-161.37-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.283 - 0.959i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.888 − 0.458i)2-s + (−0.786 − 0.618i)3-s + (0.580 + 0.814i)4-s + (−0.723 − 0.690i)5-s + (0.415 + 0.909i)6-s + (−0.142 − 0.989i)8-s + (0.235 + 0.971i)9-s + (0.327 + 0.945i)10-s + (0.888 − 0.458i)11-s + (0.0475 − 0.998i)12-s + (−0.654 + 0.755i)13-s + (0.142 + 0.989i)15-s + (−0.327 + 0.945i)16-s + (0.995 + 0.0950i)17-s + (0.235 − 0.971i)18-s + (0.995 − 0.0950i)19-s + ⋯
L(s)  = 1  + (−0.888 − 0.458i)2-s + (−0.786 − 0.618i)3-s + (0.580 + 0.814i)4-s + (−0.723 − 0.690i)5-s + (0.415 + 0.909i)6-s + (−0.142 − 0.989i)8-s + (0.235 + 0.971i)9-s + (0.327 + 0.945i)10-s + (0.888 − 0.458i)11-s + (0.0475 − 0.998i)12-s + (−0.654 + 0.755i)13-s + (0.142 + 0.989i)15-s + (−0.327 + 0.945i)16-s + (0.995 + 0.0950i)17-s + (0.235 − 0.971i)18-s + (0.995 − 0.0950i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.283 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.283 - 0.959i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.283 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4302854454 - 0.5757764878i\)
\(L(\frac12)\) \(\approx\) \(0.4302854454 - 0.5757764878i\)
\(L(1)\) \(\approx\) \(0.4952252589 - 0.2610176103i\)
\(L(1)\) \(\approx\) \(0.4952252589 - 0.2610176103i\)

Euler product

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

−27.523447190397227817864569438022, −27.12084720668276863004482164726, −26.190866557268799388767513281083, −25.04702857533332194368343386550, −23.9662577202722946548864775985, −22.86993439155390506055308133012, −22.39143946507653326783570797118, −20.8131196450841206805251617560, −19.79962231226624152754829475474, −18.80783349756561796428980991121, −17.76078139766810685961425167888, −17.00874020723341952833800801985, −15.94118037521099555544737543622, −15.15802529483726610512006712879, −14.36067407625573009235181750161, −12.0541193737018072358702249111, −11.50328232147222134074970218224, −10.20052355710616215338851322923, −9.68459761484689493925413974355, −8.05166636065196779567551355722, −7.04304535277208822868478947937, −6.008315374614327060797844564804, −4.7031928296368326475880496294, −3.11650513427669707604887076300, −0.90834441733907707592134052190, 0.58222635070738384519545734384, 1.61031269589880294188984360822, 3.50349297371994567153703018355, 5.0236748441712767480061258921, 6.64642450298525640798062978156, 7.577603754046187670672260828893, 8.63238615741259887188192503050, 9.81562161147550340133068182138, 11.20017791512450687120609729262, 11.96963654808623288462179412590, 12.49142443827099543569140542026, 14.00350109360543516343541674778, 15.847245762368566706403171951155, 16.66023298929063169319248817840, 17.2449486497694171784815692242, 18.52055828957267473755751734578, 19.30581398783066191816159977931, 20.00753446867368133346580980298, 21.347259500683415890136222170102, 22.28545348937659147047103524505, 23.54068282759384053374443227770, 24.4769154972730850753994914507, 25.139800250855197404283567392975, 26.70068020597964829461829362733, 27.42961472863411876149757483001

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