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Properties

Label 2-161-161.160-c3-0-0
Degree $2$
Conductor $161$
Sign $0.457 - 0.889i$
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  − 2-s − 5.83i·3-s − 7·4-s − 20.8·5-s + 5.83i·6-s + (−5.36 − 17.7i)7-s + 15·8-s − 7·9-s + 20.8·10-s − 15.7i·11-s + 40.8i·12-s + 35.4i·13-s + (5.36 + 17.7i)14-s + 121. i·15-s + 41·16-s + 20.8·17-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.12i·3-s − 0.875·4-s − 1.86·5-s + 0.396i·6-s + (−0.289 − 0.957i)7-s + 0.662·8-s − 0.259·9-s + 0.660·10-s − 0.432i·11-s + 0.981i·12-s + 0.755i·13-s + (0.102 + 0.338i)14-s + 2.09i·15-s + 0.640·16-s + 0.298·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.457 - 0.889i$
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: \((2,\ 161,\ (\ :3/2),\ 0.457 - 0.889i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.123645 + 0.0754193i\)
\(L(\frac12)\) \(\approx\) \(0.123645 + 0.0754193i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (5.36 + 17.7i)T \)
23 \( 1 + (-79.2 - 76.7i)T \)
good2 \( 1 + T + 8T^{2} \)
3 \( 1 + 5.83iT - 27T^{2} \)
5 \( 1 + 20.8T + 125T^{2} \)
11 \( 1 + 15.7iT - 1.33e3T^{2} \)
13 \( 1 - 35.4iT - 2.19e3T^{2} \)
17 \( 1 - 20.8T + 4.91e3T^{2} \)
19 \( 1 + 92.0T + 6.85e3T^{2} \)
29 \( 1 + 150.T + 2.43e4T^{2} \)
31 \( 1 + 172. iT - 2.97e4T^{2} \)
37 \( 1 - 69.9iT - 5.06e4T^{2} \)
41 \( 1 - 446. iT - 6.89e4T^{2} \)
43 \( 1 + 83.5iT - 7.95e4T^{2} \)
47 \( 1 + 8.86iT - 1.03e5T^{2} \)
53 \( 1 - 588. iT - 1.48e5T^{2} \)
59 \( 1 + 522. iT - 2.05e5T^{2} \)
61 \( 1 + 170.T + 2.26e5T^{2} \)
67 \( 1 + 895. iT - 3.00e5T^{2} \)
71 \( 1 - 241.T + 3.57e5T^{2} \)
73 \( 1 - 775. iT - 3.89e5T^{2} \)
79 \( 1 - 514. iT - 4.93e5T^{2} \)
83 \( 1 + 209.T + 5.71e5T^{2} \)
89 \( 1 - 1.21e3T + 7.04e5T^{2} \)
97 \( 1 + 379.T + 9.12e5T^{2} \)
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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

−12.73325526972453875352401305693, −11.63749993334397717280449517757, −10.77664477806464299382964179492, −9.310710130967167549306814605035, −8.066727129031217858452054154576, −7.63339963863715497070867790714, −6.64280829283607518514034145160, −4.50824449689643483457682610022, −3.65643574355506609155854116030, −1.01812641172274464070430000731, 0.10475180723656594614710010488, 3.38881974791101304656051919029, 4.30315103140294203612502694243, 5.21998034596156760689628180908, 7.27579216211827028271013002039, 8.453116326644554964734963502422, 8.988730493645139708556412173584, 10.22622428533802531785769460861, 10.99376964751838448986908130406, 12.30710996150054199989150989107

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