Properties

Label 1-161-161.67-r1-0-0
Degree $1$
Conductor $161$
Sign $0.714 + 0.700i$
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  + (−0.995 − 0.0950i)2-s + (0.723 + 0.690i)3-s + (0.981 + 0.189i)4-s + (0.888 − 0.458i)5-s + (−0.654 − 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (−0.928 + 0.371i)10-s + (0.995 − 0.0950i)11-s + (0.580 + 0.814i)12-s + (−0.142 + 0.989i)13-s + (0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (0.327 − 0.945i)17-s + (0.0475 − 0.998i)18-s + (0.327 + 0.945i)19-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (0.723 + 0.690i)3-s + (0.981 + 0.189i)4-s + (0.888 − 0.458i)5-s + (−0.654 − 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (−0.928 + 0.371i)10-s + (0.995 − 0.0950i)11-s + (0.580 + 0.814i)12-s + (−0.142 + 0.989i)13-s + (0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (0.327 − 0.945i)17-s + (0.0475 − 0.998i)18-s + (0.327 + 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.714 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.714 + 0.700i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.714 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.757434518 + 0.7177625577i\)
\(L(\frac12)\) \(\approx\) \(1.757434518 + 0.7177625577i\)
\(L(1)\) \(\approx\) \(1.127056423 + 0.2355427061i\)
\(L(1)\) \(\approx\) \(1.127056423 + 0.2355427061i\)

Euler product

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

−27.34023161617452370741752979711, −26.09595977676386945739316381304, −25.72726235439202299058042104955, −24.75181139875802102650959525829, −24.109776820017328927439818256339, −22.50214164371627480828097152519, −21.28290718306835557821687547898, −20.21346949765964965434339716105, −19.47658556696876849210133028029, −18.48450531828723121431343208540, −17.657800719812890051918830626408, −16.96221958906494125431958757265, −15.23116783665997508594141398298, −14.60561772804464329758137162537, −13.36924100184878939747181216737, −12.21873573962539904512370711438, −10.869265641848023356233524888276, −9.7151340561906281669568438013, −8.93523661603215974785717338302, −7.7431104200776225051474999365, −6.75396174076108712128364613881, −5.83111366917314248349978704348, −3.3036639313328352683457678366, −2.168696010633057559266635771072, −1.022724865052365175499209746424, 1.36656509777211546382245901035, 2.52805097567221832350602757038, 4.03033751933920793132153489921, 5.65576012010816774821374044279, 7.05521847670317042362541526933, 8.3932490716367804485488112192, 9.40860510255548478016675150683, 9.72419347646108267954843841633, 11.10388948481505578685578134813, 12.30343259010455985401131428010, 13.89712273432921729989197174623, 14.61611529206196436541594059959, 16.172769805368290176325595837167, 16.59587568657614306243658725701, 17.72027198333294801307486656120, 18.94256138332123131616467356664, 19.78810324889828231754490081965, 20.86035400051187289923943489927, 21.287564012939897789243656375700, 22.48266787788252809287103326414, 24.45716768195814586789806513594, 24.97456355193785833411942111602, 25.82289495022643093181327949623, 26.75013283550055487776321703522, 27.521938007656970124525696873535

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