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Properties

Label 2-161-161.160-c3-0-35
Degree $2$
Conductor $161$
Sign $-0.781 + 0.624i$
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-s − 5.83i·3-s − 7·4-s + 7.58·5-s + 5.83i·6-s + (14.7 − 11.1i)7-s + 15·8-s − 7·9-s − 7.58·10-s − 25.0i·11-s + 40.8i·12-s − 29.5i·13-s + (−14.7 + 11.1i)14-s − 44.2i·15-s + 41·16-s − 7.58·17-s + ⋯
L(s)  = 1  − 0.353·2-s − 1.12i·3-s − 0.875·4-s + 0.678·5-s + 0.396i·6-s + (0.797 − 0.603i)7-s + 0.662·8-s − 0.259·9-s − 0.239·10-s − 0.687i·11-s + 0.981i·12-s − 0.631i·13-s + (−0.282 + 0.213i)14-s − 0.760i·15-s + 0.640·16-s − 0.108·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.781 + 0.624i$
Analytic conductor: \(9.49930\)
Root analytic conductor: \(3.08209\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 161,\ (\ :3/2),\ -0.781 + 0.624i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.379898 - 1.08390i\)
\(L(\frac12)\) \(\approx\) \(0.379898 - 1.08390i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-14.7 + 11.1i)T \)
23 \( 1 + (110. - 2.96i)T \)
good2 \( 1 + T + 8T^{2} \)
3 \( 1 + 5.83iT - 27T^{2} \)
5 \( 1 - 7.58T + 125T^{2} \)
11 \( 1 + 25.0iT - 1.33e3T^{2} \)
13 \( 1 + 29.5iT - 2.19e3T^{2} \)
17 \( 1 + 7.58T + 4.91e3T^{2} \)
19 \( 1 + 146.T + 6.85e3T^{2} \)
29 \( 1 - 228.T + 2.43e4T^{2} \)
31 \( 1 - 282. iT - 2.97e4T^{2} \)
37 \( 1 + 364. iT - 5.06e4T^{2} \)
41 \( 1 + 73.4iT - 6.89e4T^{2} \)
43 \( 1 + 370. iT - 7.95e4T^{2} \)
47 \( 1 - 446. iT - 1.03e5T^{2} \)
53 \( 1 + 490. iT - 1.48e5T^{2} \)
59 \( 1 + 2.36iT - 2.05e5T^{2} \)
61 \( 1 - 421.T + 2.26e5T^{2} \)
67 \( 1 - 478. iT - 3.00e5T^{2} \)
71 \( 1 - 620.T + 3.57e5T^{2} \)
73 \( 1 + 134. iT - 3.89e5T^{2} \)
79 \( 1 - 1.29e3iT - 4.93e5T^{2} \)
83 \( 1 + 821.T + 5.71e5T^{2} \)
89 \( 1 - 97.4T + 7.04e5T^{2} \)
97 \( 1 - 1.39e3T + 9.12e5T^{2} \)
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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

−12.33862248920123310847555578057, −10.79905969647407689798303167170, −10.08458038704906927316084298601, −8.581470173010334227788457874622, −8.061552177203334616155711141810, −6.81275691735104844488437721917, −5.57099409432615680542016623312, −4.15631520210194553220164136800, −1.93479918577275685230102708085, −0.63990628196875141226608962951, 1.96811695962049755669807488703, 4.24997533619553605580864432364, 4.77610722855011541814423209899, 6.17186621393194177513158866107, 8.041429475627216114851458833227, 8.926001819686495256882086729366, 9.819538649166059297100852652676, 10.34652672675092762972476892521, 11.64281006777692038913448434832, 12.87913897970129037625920014608

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