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Properties

Label 2-161-7.2-c3-0-16
Degree $2$
Conductor $161$
Sign $-0.982 - 0.187i$
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  + (−0.702 + 1.21i)2-s + (4.90 + 8.48i)3-s + (3.01 + 5.21i)4-s + (−2.29 + 3.98i)5-s − 13.7·6-s + (4.61 − 17.9i)7-s − 19.7·8-s + (−34.5 + 59.8i)9-s + (−3.23 − 5.59i)10-s + (17.8 + 30.9i)11-s + (−29.5 + 51.1i)12-s + 36.1·13-s + (18.5 + 18.2i)14-s − 45.0·15-s + (−10.2 + 17.7i)16-s + (−35.4 − 61.3i)17-s + ⋯
L(s)  = 1  + (−0.248 + 0.430i)2-s + (0.943 + 1.63i)3-s + (0.376 + 0.652i)4-s + (−0.205 + 0.356i)5-s − 0.937·6-s + (0.249 − 0.968i)7-s − 0.871·8-s + (−1.27 + 2.21i)9-s + (−0.102 − 0.176i)10-s + (0.489 + 0.848i)11-s + (−0.710 + 1.22i)12-s + 0.770·13-s + (0.355 + 0.347i)14-s − 0.775·15-s + (−0.159 + 0.276i)16-s + (−0.505 − 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.982 - 0.187i$
Analytic conductor: \(9.49930\)
Root analytic conductor: \(3.08209\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :3/2),\ -0.982 - 0.187i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.194603 + 2.06011i\)
\(L(\frac12)\) \(\approx\) \(0.194603 + 2.06011i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-4.61 + 17.9i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good2 \( 1 + (0.702 - 1.21i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-4.90 - 8.48i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (2.29 - 3.98i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-17.8 - 30.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 36.1T + 2.19e3T^{2} \)
17 \( 1 + (35.4 + 61.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-62.2 + 107. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 + 55.6T + 2.43e4T^{2} \)
31 \( 1 + (-111. - 192. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-216. + 374. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 314.T + 6.89e4T^{2} \)
43 \( 1 + 443.T + 7.95e4T^{2} \)
47 \( 1 + (216. - 375. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-85.9 - 148. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (84.2 + 145. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (10.2 - 17.8i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-276. - 478. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 553.T + 3.57e5T^{2} \)
73 \( 1 + (-27.4 - 47.4i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-295. + 512. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 422.T + 5.71e5T^{2} \)
89 \( 1 + (344. - 596. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 628.T + 9.12e5T^{2} \)
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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

−13.16760474290222861848100412515, −11.37928475295827853659939126257, −10.88629152483590681253119177607, −9.566493198459538422698903263583, −8.932408067978942523225643568206, −7.76137725217529581288572343965, −6.90955215893846438949392271414, −4.82344755124741399471279305004, −3.82093042781026192612130861960, −2.81039609482325114061859149666, 1.00894761224540728581510512007, 1.98690767074208025162573839409, 3.28705376541963404581237381023, 5.90672904788851580003522647966, 6.46164085834354941138518205026, 8.151745441916994720935557898967, 8.550187858974698726455642093149, 9.656590752873331529716550571231, 11.39424892554016267980723701666, 11.89570988110199357732407462863

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