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Properties

Label 2-161-7.4-c3-0-33
Degree $2$
Conductor $161$
Sign $0.985 + 0.169i$
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  + (2.08 + 3.61i)2-s + (3.80 − 6.59i)3-s + (−4.70 + 8.15i)4-s + (−1.49 − 2.59i)5-s + 31.8·6-s + (1.97 − 18.4i)7-s − 5.91·8-s + (−15.5 − 26.9i)9-s + (6.24 − 10.8i)10-s + (27.0 − 46.7i)11-s + (35.8 + 62.1i)12-s − 50.0·13-s + (70.6 − 31.2i)14-s − 22.7·15-s + (25.3 + 43.8i)16-s + (−16.4 + 28.4i)17-s + ⋯
L(s)  = 1  + (0.737 + 1.27i)2-s + (0.733 − 1.26i)3-s + (−0.588 + 1.01i)4-s + (−0.133 − 0.231i)5-s + 2.16·6-s + (0.106 − 0.994i)7-s − 0.261·8-s + (−0.575 − 0.996i)9-s + (0.197 − 0.341i)10-s + (0.740 − 1.28i)11-s + (0.863 + 1.49i)12-s − 1.06·13-s + (1.34 − 0.597i)14-s − 0.392·15-s + (0.395 + 0.685i)16-s + (−0.234 + 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.985 + 0.169i$
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: \((2,\ 161,\ (\ :3/2),\ 0.985 + 0.169i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.06510 - 0.261821i\)
\(L(\frac12)\) \(\approx\) \(3.06510 - 0.261821i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.97 + 18.4i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good2 \( 1 + (-2.08 - 3.61i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-3.80 + 6.59i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (1.49 + 2.59i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-27.0 + 46.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 50.0T + 2.19e3T^{2} \)
17 \( 1 + (16.4 - 28.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-40.6 - 70.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 244.T + 2.43e4T^{2} \)
31 \( 1 + (90.7 - 157. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-131. - 227. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 442.T + 6.89e4T^{2} \)
43 \( 1 + 194.T + 7.95e4T^{2} \)
47 \( 1 + (-167. - 289. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-139. + 240. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (43.8 - 75.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (97.4 + 168. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-304. + 526. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 890.T + 3.57e5T^{2} \)
73 \( 1 + (250. - 433. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-323. - 560. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 94.1T + 5.71e5T^{2} \)
89 \( 1 + (-325. - 563. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 968.T + 9.12e5T^{2} \)
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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

−12.81740959460830774355541995192, −11.90826085016943756390164926448, −10.32202127510183059701123287696, −8.545710490812626988200113512678, −7.972180031900628505161106505258, −6.97533280217612585389058364310, −6.33861564022398046587583935177, −4.78891836134948334685263736394, −3.37197412523759804357971265840, −1.21995430252630831343713051233, 2.23082636346589063586961494726, 3.13522292194602744834523954897, 4.43794807541020757959538469419, 5.06025595759102783140474927124, 7.21226682674082754145299992873, 8.944622023780925961520543642654, 9.632543087765998692411576436615, 10.39240452562517051669601557708, 11.62889525425318494011236629271, 12.16479081425769103923966825800

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