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Properties

Label 1-161-161.65-r1-0-0
Degree $1$
Conductor $161$
Sign $-0.993 + 0.113i$
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  + (0.981 + 0.189i)2-s + (0.0475 + 0.998i)3-s + (0.928 + 0.371i)4-s + (−0.580 + 0.814i)5-s + (−0.142 + 0.989i)6-s + (0.841 + 0.540i)8-s + (−0.995 + 0.0950i)9-s + (−0.723 + 0.690i)10-s + (−0.981 + 0.189i)11-s + (−0.327 + 0.945i)12-s + (−0.959 − 0.281i)13-s + (−0.841 − 0.540i)15-s + (0.723 + 0.690i)16-s + (0.786 + 0.618i)17-s + (−0.995 − 0.0950i)18-s + (0.786 − 0.618i)19-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)2-s + (0.0475 + 0.998i)3-s + (0.928 + 0.371i)4-s + (−0.580 + 0.814i)5-s + (−0.142 + 0.989i)6-s + (0.841 + 0.540i)8-s + (−0.995 + 0.0950i)9-s + (−0.723 + 0.690i)10-s + (−0.981 + 0.189i)11-s + (−0.327 + 0.945i)12-s + (−0.959 − 0.281i)13-s + (−0.841 − 0.540i)15-s + (0.723 + 0.690i)16-s + (0.786 + 0.618i)17-s + (−0.995 − 0.0950i)18-s + (0.786 − 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.993 + 0.113i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ -0.993 + 0.113i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1286166954 + 2.259124498i\)
\(L(\frac12)\) \(\approx\) \(0.1286166954 + 2.259124498i\)
\(L(1)\) \(\approx\) \(1.149852035 + 1.059908392i\)
\(L(1)\) \(\approx\) \(1.149852035 + 1.059908392i\)

Euler product

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

−27.18143318032736751894760264669, −25.714662333309889333408978680945, −24.69521933453901832159883240437, −24.1111235760615485367197792905, −23.34588856519577624086467979175, −22.48852652273790522550250250210, −21.03688673503951005441522462106, −20.31232700821136002205295227670, −19.37093905107045784738070422788, −18.49948952115271806583046692239, −16.91366915107040771238295072467, −16.06812875330883021287524948243, −14.815393353623143932383567719956, −13.77501598601915796458706204014, −12.852743825720296855787702117755, −12.11265534898967782884012277480, −11.32902321008866597603329177310, −9.68295088387825681322872341992, −7.92625678186161869100595498088, −7.35578835041324876168614680154, −5.7777638322521358579368578631, −4.92345794080710443806390677786, −3.360480964892216509055082870819, −2.07109584892192718742041378887, −0.56505373259457305156321334059, 2.66027749186395053669010145607, 3.43489931745385968526370185980, 4.714008077033374848283777964868, 5.62782398467944199881282360719, 7.15502037640361053134493456987, 8.085004106757214695325048976707, 9.94591601654422744959306091450, 10.81273491537944823950621652565, 11.74392936168429593687370096677, 12.94892398446065349418572131191, 14.417109511325166026717012128341, 14.900797617877706571554058159852, 15.81102046631104094875785926418, 16.643712641419694038093277754763, 18.03595353537289495670173680817, 19.61840050311966449493592119588, 20.310201886967781669058528452859, 21.58503569762370580341251151472, 22.04214921470539920701038479632, 23.1110746702147964395648267388, 23.73667138241994469862695171790, 25.18920873663127993372010579470, 26.16808173806884845668847417612, 26.75557284058245298322485405792, 27.99721713967388056305519032699

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