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Properties

Label 2-161-7.4-c3-0-12
Degree $2$
Conductor $161$
Sign $-0.998 + 0.0620i$
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  + (1.15 + 2.00i)2-s + (−4.37 + 7.57i)3-s + (1.32 − 2.28i)4-s + (7.70 + 13.3i)5-s − 20.2·6-s + (2.31 + 18.3i)7-s + 24.6·8-s + (−24.7 − 42.8i)9-s + (−17.8 + 30.9i)10-s + (−24.3 + 42.2i)11-s + (11.5 + 19.9i)12-s + 53.5·13-s + (−34.1 + 25.9i)14-s − 134.·15-s + (17.9 + 31.0i)16-s + (42.8 − 74.3i)17-s + ⋯
L(s)  = 1  + (0.409 + 0.708i)2-s + (−0.841 + 1.45i)3-s + (0.165 − 0.285i)4-s + (0.689 + 1.19i)5-s − 1.37·6-s + (0.125 + 0.992i)7-s + 1.08·8-s + (−0.915 − 1.58i)9-s + (−0.564 + 0.977i)10-s + (−0.668 + 1.15i)11-s + (0.277 + 0.480i)12-s + 1.14·13-s + (−0.652 + 0.494i)14-s − 2.32·15-s + (0.280 + 0.485i)16-s + (0.612 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0620i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.998 + 0.0620i$
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: \((2,\ 161,\ (\ :3/2),\ -0.998 + 0.0620i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0594771 - 1.91498i\)
\(L(\frac12)\) \(\approx\) \(0.0594771 - 1.91498i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.31 - 18.3i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good2 \( 1 + (-1.15 - 2.00i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (4.37 - 7.57i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-7.70 - 13.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (24.3 - 42.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 53.5T + 2.19e3T^{2} \)
17 \( 1 + (-42.8 + 74.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (64.1 + 111. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 + 133.T + 2.43e4T^{2} \)
31 \( 1 + (-41.4 + 71.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (27.1 + 47.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 475.T + 6.89e4T^{2} \)
43 \( 1 + 146.T + 7.95e4T^{2} \)
47 \( 1 + (-261. - 453. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-135. + 235. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (87.0 - 150. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-225. - 389. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (297. - 515. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 438.T + 3.57e5T^{2} \)
73 \( 1 + (300. - 520. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (435. + 755. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 229.T + 5.71e5T^{2} \)
89 \( 1 + (501. + 868. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.55e3T + 9.12e5T^{2} \)
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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

−13.11023959510334433672216845043, −11.45854117721217493278345698703, −10.86438163870728398642358072515, −10.05958089755585207204374564864, −9.223933552137216810980496440032, −7.24320520928646031003741915670, −6.11106795951858242435189952912, −5.48939597669704843438287145723, −4.46829978885398160739982867814, −2.57819913427835631728047272779, 0.940729231782780909445014866902, 1.75305631700973701653147815684, 3.81823586870699581931226354216, 5.46459343831939300121000614412, 6.30723993109545275175380568943, 7.79101405133463191581034000068, 8.401771954803520521069399783689, 10.49909816989197243721550633467, 11.04589278159378544614560295446, 12.22199167858069696057840394272

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