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Properties

Label 2-161-161.160-c3-0-45
Degree $2$
Conductor $161$
Sign $-0.802 - 0.596i$
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  + 1.68·2-s − 8.88i·3-s − 5.16·4-s + 0.217·5-s − 14.9i·6-s + (−7.49 + 16.9i)7-s − 22.1·8-s − 51.9·9-s + 0.367·10-s − 27.1i·11-s + 45.8i·12-s + 13.3i·13-s + (−12.6 + 28.5i)14-s − 1.93i·15-s + 3.94·16-s + 84.1·17-s + ⋯
L(s)  = 1  + 0.595·2-s − 1.71i·3-s − 0.645·4-s + 0.0194·5-s − 1.01i·6-s + (−0.404 + 0.914i)7-s − 0.979·8-s − 1.92·9-s + 0.0116·10-s − 0.743i·11-s + 1.10i·12-s + 0.284i·13-s + (−0.241 + 0.544i)14-s − 0.0333i·15-s + 0.0615·16-s + 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.802 - 0.596i$
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: \((2,\ 161,\ (\ :3/2),\ -0.802 - 0.596i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.155211 + 0.469103i\)
\(L(\frac12)\) \(\approx\) \(0.155211 + 0.469103i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (7.49 - 16.9i)T \)
23 \( 1 + (24.3 - 107. i)T \)
good2 \( 1 - 1.68T + 8T^{2} \)
3 \( 1 + 8.88iT - 27T^{2} \)
5 \( 1 - 0.217T + 125T^{2} \)
11 \( 1 + 27.1iT - 1.33e3T^{2} \)
13 \( 1 - 13.3iT - 2.19e3T^{2} \)
17 \( 1 - 84.1T + 4.91e3T^{2} \)
19 \( 1 + 126.T + 6.85e3T^{2} \)
29 \( 1 + 187.T + 2.43e4T^{2} \)
31 \( 1 + 163. iT - 2.97e4T^{2} \)
37 \( 1 + 347. iT - 5.06e4T^{2} \)
41 \( 1 + 395. iT - 6.89e4T^{2} \)
43 \( 1 - 340. iT - 7.95e4T^{2} \)
47 \( 1 + 200. iT - 1.03e5T^{2} \)
53 \( 1 - 40.1iT - 1.48e5T^{2} \)
59 \( 1 + 216. iT - 2.05e5T^{2} \)
61 \( 1 + 909.T + 2.26e5T^{2} \)
67 \( 1 - 383. iT - 3.00e5T^{2} \)
71 \( 1 - 688.T + 3.57e5T^{2} \)
73 \( 1 - 145. iT - 3.89e5T^{2} \)
79 \( 1 + 150. iT - 4.93e5T^{2} \)
83 \( 1 - 613.T + 5.71e5T^{2} \)
89 \( 1 + 473.T + 7.04e5T^{2} \)
97 \( 1 - 541.T + 9.12e5T^{2} \)
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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

−12.25669337733239340215091510557, −11.35004037361029402120485003267, −9.471166756719970322255554217491, −8.533370730061242833051290396018, −7.57119543035527345389739822040, −6.04690009308790481226289410674, −5.70401758656422524277763838734, −3.59445213868948300149506034076, −2.10370318052873706616839054883, −0.18639413757463850669518006278, 3.30259039178005425939698486890, 4.18496686601830907367632042921, 4.93530458236498107767010608847, 6.20846367360647183740094100615, 8.072536397672193116550627819277, 9.286128226416395737303981190579, 10.04671813133624916609349528209, 10.65117779018289106415081376294, 12.11062738344847293655990624251, 13.11535380602705792055788471200

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