Properties

Label 1-161-161.100-r0-0-0
Degree $1$
Conductor $161$
Sign $0.972 - 0.230i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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  + (0.235 + 0.971i)2-s + (−0.327 − 0.945i)3-s + (−0.888 + 0.458i)4-s + (0.928 − 0.371i)5-s + (0.841 − 0.540i)6-s + (−0.654 − 0.755i)8-s + (−0.786 + 0.618i)9-s + (0.580 + 0.814i)10-s + (0.235 − 0.971i)11-s + (0.723 + 0.690i)12-s + (0.415 − 0.909i)13-s + (−0.654 − 0.755i)15-s + (0.580 − 0.814i)16-s + (0.0475 + 0.998i)17-s + (−0.786 − 0.618i)18-s + (0.0475 − 0.998i)19-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)2-s + (−0.327 − 0.945i)3-s + (−0.888 + 0.458i)4-s + (0.928 − 0.371i)5-s + (0.841 − 0.540i)6-s + (−0.654 − 0.755i)8-s + (−0.786 + 0.618i)9-s + (0.580 + 0.814i)10-s + (0.235 − 0.971i)11-s + (0.723 + 0.690i)12-s + (0.415 − 0.909i)13-s + (−0.654 − 0.755i)15-s + (0.580 − 0.814i)16-s + (0.0475 + 0.998i)17-s + (−0.786 − 0.618i)18-s + (0.0475 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.972 - 0.230i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.972 - 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.140464667 - 0.1334960472i\)
\(L(\frac12)\) \(\approx\) \(1.140464667 - 0.1334960472i\)
\(L(1)\) \(\approx\) \(1.098104372 + 0.04833142605i\)
\(L(1)\) \(\approx\) \(1.098104372 + 0.04833142605i\)

Euler product

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

−28.01127077089784088127251567393, −27.04511119520656601727550066973, −26.169627389639424180027909190918, −25.09514566023665532204420124797, −23.40117299134288785945176849325, −22.66181256414612539603994687034, −21.89528589726200538974123058171, −20.93496610922087735650110003480, −20.48630439506590288180939113221, −19.01011377761971914075312960959, −17.9448182384246398951720375732, −17.17591162348650734171249741478, −15.817415146301679761096146587411, −14.4296300364818953635996947704, −13.97088631199231778390333195779, −12.426106242559781923116604878787, −11.48776631962617383593928136018, −10.36187108530527975985701516084, −9.729678496264360485977920590623, −8.83876409858315497446108143199, −6.60441973567662915982454092739, −5.37160691467321430635597954293, −4.390661031799421617269343491028, −3.120711946672255845032621615806, −1.74338732179784294003021478215, 1.05741005514012727061212386085, 2.99088129318944575768163276970, 4.91094907666088991981441164237, 5.98448120034902219225388531459, 6.52756885407699678339155680557, 8.07898879179938935213559689608, 8.73497313109399643451033115624, 10.29265265933411448319824959287, 11.835621196749981744149677114581, 13.2201704539695389450718083341, 13.37229539622581128198619163907, 14.59231445968533014826894312469, 15.95194358464693343303117978046, 17.07947520791209797235076510395, 17.576097886777916774876125236923, 18.53133968231888724405277345602, 19.67172243517104142050575388809, 21.267644983104450633482394329425, 22.105961692837768665025707309899, 23.125953695886093178877020551717, 24.13349697399438745651464932052, 24.716383813738132695553055611779, 25.52276083220076940179707355654, 26.42076110814152526714843120164, 27.84646290696294912340284122577

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