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Properties

Label 2-161-7.2-c3-0-8
Degree $2$
Conductor $161$
Sign $0.942 + 0.334i$
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  + (−2.21 + 3.83i)2-s + (−4.64 − 8.03i)3-s + (−5.80 − 10.0i)4-s + (−8.69 + 15.0i)5-s + 41.0·6-s + (−7.29 + 17.0i)7-s + 15.9·8-s + (−29.5 + 51.2i)9-s + (−38.5 − 66.7i)10-s + (−5.80 − 10.0i)11-s + (−53.8 + 93.2i)12-s − 43.4·13-s + (−49.1 − 65.6i)14-s + 161.·15-s + (11.1 − 19.2i)16-s + (27.5 + 47.7i)17-s + ⋯
L(s)  = 1  + (−0.782 + 1.35i)2-s + (−0.893 − 1.54i)3-s + (−0.725 − 1.25i)4-s + (−0.778 + 1.34i)5-s + 2.79·6-s + (−0.393 + 0.919i)7-s + 0.704·8-s + (−1.09 + 1.89i)9-s + (−1.21 − 2.10i)10-s + (−0.158 − 0.275i)11-s + (−1.29 + 2.24i)12-s − 0.926·13-s + (−0.938 − 1.25i)14-s + 2.78·15-s + (0.173 − 0.300i)16-s + (0.393 + 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.942 + 0.334i$
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: \((2,\ 161,\ (\ :3/2),\ 0.942 + 0.334i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.223200 - 0.0384518i\)
\(L(\frac12)\) \(\approx\) \(0.223200 - 0.0384518i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (7.29 - 17.0i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good2 \( 1 + (2.21 - 3.83i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (4.64 + 8.03i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (8.69 - 15.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (5.80 + 10.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 43.4T + 2.19e3T^{2} \)
17 \( 1 + (-27.5 - 47.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-40.3 + 69.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 + 55.6T + 2.43e4T^{2} \)
31 \( 1 + (120. + 208. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (121. - 211. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 257.T + 6.89e4T^{2} \)
43 \( 1 - 142.T + 7.95e4T^{2} \)
47 \( 1 + (-321. + 556. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-241. - 418. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-175. - 303. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (172. - 298. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-377. - 654. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 877.T + 3.57e5T^{2} \)
73 \( 1 + (612. + 1.06e3i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (110. - 190. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 416.T + 5.71e5T^{2} \)
89 \( 1 + (-770. + 1.33e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 754.T + 9.12e5T^{2} \)
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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

−12.17150095083762564994086117471, −11.55339637411459652915213319512, −10.34449039277472381432251228107, −8.799656126933134184543522073944, −7.52973066828670495093712781734, −7.28455402369629245581666779020, −6.26910529537574978423923687827, −5.55899991185878296896945613507, −2.64732505038356379379760938663, −0.23526061101613678886231965893, 0.799186309028627392836823046450, 3.49010082805842236499957678395, 4.34872753086072239081410147626, 5.37340996747443438678375571896, 7.63246723690305675268177024860, 9.084188363252221092464312487064, 9.625191237087572855826264394883, 10.44224424145943689411889774344, 11.25781372732219325274850073468, 12.18582227174785526322738523460

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