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Properties

Label 2-161-7.4-c3-0-4
Degree $2$
Conductor $161$
Sign $0.279 + 0.960i$
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  + (2.42 + 4.19i)2-s + (−4.52 + 7.83i)3-s + (−7.73 + 13.3i)4-s + (−7.08 − 12.2i)5-s − 43.8·6-s + (17.4 − 6.30i)7-s − 36.1·8-s + (−27.4 − 47.5i)9-s + (34.2 − 59.4i)10-s + (−24.1 + 41.8i)11-s + (−69.9 − 121. i)12-s − 21.7·13-s + (68.6 + 57.7i)14-s + 128.·15-s + (−25.6 − 44.4i)16-s + (−54.5 + 94.5i)17-s + ⋯
L(s)  = 1  + (0.856 + 1.48i)2-s + (−0.870 + 1.50i)3-s + (−0.966 + 1.67i)4-s + (−0.633 − 1.09i)5-s − 2.98·6-s + (0.940 − 0.340i)7-s − 1.59·8-s + (−1.01 − 1.75i)9-s + (1.08 − 1.87i)10-s + (−0.661 + 1.14i)11-s + (−1.68 − 2.91i)12-s − 0.464·13-s + (1.30 + 1.10i)14-s + 2.20·15-s + (−0.401 − 0.694i)16-s + (−0.778 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.279 + 0.960i$
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: \((2,\ 161,\ (\ :3/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.653339 - 0.490026i\)
\(L(\frac12)\) \(\approx\) \(0.653339 - 0.490026i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-17.4 + 6.30i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good2 \( 1 + (-2.42 - 4.19i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (4.52 - 7.83i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (7.08 + 12.2i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (24.1 - 41.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 21.7T + 2.19e3T^{2} \)
17 \( 1 + (54.5 - 94.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (59.9 + 103. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 85.5T + 2.43e4T^{2} \)
31 \( 1 + (-36.8 + 63.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-127. - 220. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 108.T + 6.89e4T^{2} \)
43 \( 1 + 447.T + 7.95e4T^{2} \)
47 \( 1 + (-25.3 - 43.8i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (310. - 538. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-234. + 406. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (87.0 + 150. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (138. - 239. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 77.7T + 3.57e5T^{2} \)
73 \( 1 + (433. - 750. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-197. - 342. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 207.T + 5.71e5T^{2} \)
89 \( 1 + (-494. - 856. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.13e3T + 9.12e5T^{2} \)
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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

−13.24387976961324709628834258082, −12.39392607398430717951566130140, −11.31363901566550549101054800499, −10.20645367643904643034023959383, −8.798235302636033032717058771793, −7.971033605132359105381519440996, −6.58622772076960866956139131054, −5.15064927672222944332523774315, −4.64909306679849075586810683922, −4.19652640848887627645793995794, 0.31578639625973095296821880056, 1.96147358028849628302844177379, 3.01491626670708281082775994472, 4.85555270286815958624898500393, 5.94352884051044092382264916473, 7.20262312604261990252893198156, 8.288087698340973419873347331220, 10.43802169213185799753571741648, 11.18187515503977593521293637386, 11.61006625232740220536765120100

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