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Properties

Label 2-161-7.4-c3-0-43
Degree $2$
Conductor $161$
Sign $0.382 - 0.924i$
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  + (−2.44 − 4.23i)2-s + (3.42 − 5.92i)3-s + (−7.98 + 13.8i)4-s + (−9.57 − 16.5i)5-s − 33.4·6-s + (−17.5 − 5.98i)7-s + 38.9·8-s + (−9.89 − 17.1i)9-s + (−46.8 + 81.1i)10-s + (27.4 − 47.5i)11-s + (54.5 + 94.5i)12-s + 53.1·13-s + (17.5 + 88.9i)14-s − 130.·15-s + (−31.5 − 54.5i)16-s + (−13.7 + 23.7i)17-s + ⋯
L(s)  = 1  + (−0.865 − 1.49i)2-s + (0.658 − 1.14i)3-s + (−0.997 + 1.72i)4-s + (−0.856 − 1.48i)5-s − 2.27·6-s + (−0.946 − 0.322i)7-s + 1.72·8-s + (−0.366 − 0.634i)9-s + (−1.48 + 2.56i)10-s + (0.751 − 1.30i)11-s + (1.31 + 2.27i)12-s + 1.13·13-s + (0.334 + 1.69i)14-s − 2.25·15-s + (−0.492 − 0.853i)16-s + (−0.195 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.382 - 0.924i$
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: \((2,\ 161,\ (\ :3/2),\ 0.382 - 0.924i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.687489 + 0.459614i\)
\(L(\frac12)\) \(\approx\) \(0.687489 + 0.459614i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (17.5 + 5.98i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good2 \( 1 + (2.44 + 4.23i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-3.42 + 5.92i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (9.57 + 16.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-27.4 + 47.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 53.1T + 2.19e3T^{2} \)
17 \( 1 + (13.7 - 23.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (43.6 + 75.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 285.T + 2.43e4T^{2} \)
31 \( 1 + (-28.1 + 48.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (8.63 + 14.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 11.3T + 6.89e4T^{2} \)
43 \( 1 + 155.T + 7.95e4T^{2} \)
47 \( 1 + (67.9 + 117. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (150. - 261. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-85.7 + 148. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (245. + 424. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (510. - 884. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 32.0T + 3.57e5T^{2} \)
73 \( 1 + (-359. + 622. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-549. - 952. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 433.T + 5.71e5T^{2} \)
89 \( 1 + (3.26 + 5.65i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 700.T + 9.12e5T^{2} \)
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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

−11.74613719181295690391649728970, −10.76476509944998198110456651233, −9.226566173866518862725235134617, −8.589707733135364975206433319248, −8.131790402057496220964752440258, −6.51590746299780301302324221933, −4.10589545057354926667568582039, −3.07327970147029831147329531047, −1.26764670287572808557546373062, −0.56391831399791762728735765070, 3.20879462030168729458170596729, 4.35963008619190097644236972209, 6.35850506033762738252124567631, 6.85210944572084519545884125895, 8.111572690322306569009070161900, 9.036291772860080062129189924180, 9.972874355222017008513426278837, 10.52129723771774873578026710410, 12.08768564262383500078905660378, 13.97646229583503327351944255517

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