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Properties

Label 2-161-161.160-c3-0-20
Degree $2$
Conductor $161$
Sign $0.922 + 0.387i$
Analytic cond. $9.49930$
Root an. cond. $3.08209$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.42·2-s − 3.10i·3-s + 11.5·4-s + 13.0·5-s + 13.7i·6-s + (12.3 − 13.8i)7-s − 15.8·8-s + 17.3·9-s − 57.7·10-s + 18.4i·11-s − 35.9i·12-s + 72.2i·13-s + (−54.4 + 61.2i)14-s − 40.5i·15-s − 22.4·16-s + 0.439·17-s + ⋯
L(s)  = 1  − 1.56·2-s − 0.597i·3-s + 1.44·4-s + 1.16·5-s + 0.935i·6-s + (0.664 − 0.747i)7-s − 0.700·8-s + 0.642·9-s − 1.82·10-s + 0.505i·11-s − 0.865i·12-s + 1.54i·13-s + (−1.03 + 1.16i)14-s − 0.697i·15-s − 0.351·16-s + 0.00627·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.922 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.922 + 0.387i$
Analytic conductor: \(9.49930\)
Root analytic conductor: \(3.08209\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :3/2),\ 0.922 + 0.387i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.09968 - 0.221508i\)
\(L(\frac12)\) \(\approx\) \(1.09968 - 0.221508i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-12.3 + 13.8i)T \)
23 \( 1 + (-35.6 - 104. i)T \)
good2 \( 1 + 4.42T + 8T^{2} \)
3 \( 1 + 3.10iT - 27T^{2} \)
5 \( 1 - 13.0T + 125T^{2} \)
11 \( 1 - 18.4iT - 1.33e3T^{2} \)
13 \( 1 - 72.2iT - 2.19e3T^{2} \)
17 \( 1 - 0.439T + 4.91e3T^{2} \)
19 \( 1 - 38.3T + 6.85e3T^{2} \)
29 \( 1 - 235.T + 2.43e4T^{2} \)
31 \( 1 + 84.9iT - 2.97e4T^{2} \)
37 \( 1 - 357. iT - 5.06e4T^{2} \)
41 \( 1 + 147. iT - 6.89e4T^{2} \)
43 \( 1 + 44.0iT - 7.95e4T^{2} \)
47 \( 1 + 14.3iT - 1.03e5T^{2} \)
53 \( 1 + 141. iT - 1.48e5T^{2} \)
59 \( 1 + 536. iT - 2.05e5T^{2} \)
61 \( 1 + 844.T + 2.26e5T^{2} \)
67 \( 1 + 683. iT - 3.00e5T^{2} \)
71 \( 1 - 468.T + 3.57e5T^{2} \)
73 \( 1 - 686. iT - 3.89e5T^{2} \)
79 \( 1 - 642. iT - 4.93e5T^{2} \)
83 \( 1 - 33.3T + 5.71e5T^{2} \)
89 \( 1 - 1.16e3T + 7.04e5T^{2} \)
97 \( 1 + 1.32e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.07606448552792014316590518111, −11.05117721966129895678086381301, −9.926550966479595961345185083904, −9.529034617413309614158682119323, −8.231703401083342400780730685079, −7.19604075586583803556929491720, −6.55308079898897870840652726300, −4.63823889722186169508188666212, −1.95279844626616693532581709342, −1.27280814964744971538494179801, 1.15038009066758158648320567670, 2.63278179665222208314291915268, 4.97364879205946976828963154300, 6.13860310515936850534962174302, 7.63269384335704893964693483512, 8.651018687932998857172800037969, 9.406272614777470113544037680930, 10.37886762586146325223721659958, 10.75049117340440668051896116626, 12.25107169123700841419739093176

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