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Properties

Label 2-161-7.4-c3-0-11
Degree $2$
Conductor $161$
Sign $-0.864 - 0.502i$
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  + (1.81 + 3.13i)2-s + (−0.112 + 0.195i)3-s + (−2.56 + 4.44i)4-s + (−1.55 − 2.69i)5-s − 0.817·6-s + (−8.28 + 16.5i)7-s + 10.3·8-s + (13.4 + 23.3i)9-s + (5.64 − 9.77i)10-s + (−13.6 + 23.6i)11-s + (−0.579 − 1.00i)12-s − 46.9·13-s + (−66.9 + 4.02i)14-s + 0.702·15-s + (39.3 + 68.1i)16-s + (−30.3 + 52.6i)17-s + ⋯
L(s)  = 1  + (0.640 + 1.10i)2-s + (−0.0217 + 0.0376i)3-s + (−0.320 + 0.555i)4-s + (−0.139 − 0.241i)5-s − 0.0556·6-s + (−0.447 + 0.894i)7-s + 0.459·8-s + (0.499 + 0.864i)9-s + (0.178 − 0.308i)10-s + (−0.374 + 0.648i)11-s + (−0.0139 − 0.0241i)12-s − 1.00·13-s + (−1.27 + 0.0769i)14-s + 0.0120·15-s + (0.614 + 1.06i)16-s + (−0.433 + 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.864 - 0.502i$
Analytic conductor: \(9.49930\)
Root analytic conductor: \(3.08209\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :3/2),\ -0.864 - 0.502i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.529194 + 1.96199i\)
\(L(\frac12)\) \(\approx\) \(0.529194 + 1.96199i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (8.28 - 16.5i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good2 \( 1 + (-1.81 - 3.13i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (0.112 - 0.195i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (1.55 + 2.69i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (13.6 - 23.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 46.9T + 2.19e3T^{2} \)
17 \( 1 + (30.3 - 52.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (0.916 + 1.58i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 115.T + 2.43e4T^{2} \)
31 \( 1 + (-32.9 + 57.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (4.26 + 7.38i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 129.T + 6.89e4T^{2} \)
43 \( 1 - 200.T + 7.95e4T^{2} \)
47 \( 1 + (-38.8 - 67.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-14.2 + 24.6i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (25.6 - 44.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-30.5 - 52.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-8.64 + 14.9i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 366.T + 3.57e5T^{2} \)
73 \( 1 + (-455. + 788. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-611. - 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 825.T + 5.71e5T^{2} \)
89 \( 1 + (348. + 604. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 658.T + 9.12e5T^{2} \)
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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

−12.87416676149259432609094433513, −12.33115433582182043296008821400, −10.73800243837981416363099749909, −9.753748314799284560336951920450, −8.349172650189480176984421210863, −7.42467980597244703581058791379, −6.37350196119753679946133759786, −5.20517564707635295621585456862, −4.41931712490504885616502861771, −2.27029702687444827053645783945, 0.792845145145772000889433210846, 2.74849234956810982933598187025, 3.76918875129613470907039213603, 4.91120153520309800844354696173, 6.70424173718109941940030815110, 7.59275909589948790080300945082, 9.348687653089069348374132590516, 10.28680381978303728249281129481, 11.07909636723422349360340240060, 12.08541874580112826017010394788

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