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Properties

Label 2-160-5.4-c3-0-5
Degree $2$
Conductor $160$
Sign $0.640 - 0.767i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
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  − 4.30i·3-s + (8.58 + 7.16i)5-s + 28.3i·7-s + 8.49·9-s − 65.2·11-s + 33.6i·13-s + (30.8 − 36.9i)15-s + 73.3i·17-s + 134.·19-s + 121.·21-s + 14.7i·23-s + (22.3 + 122. i)25-s − 152. i·27-s + 224.·29-s − 68.8·31-s + ⋯
L(s)  = 1  − 0.827i·3-s + (0.767 + 0.640i)5-s + 1.52i·7-s + 0.314·9-s − 1.78·11-s + 0.718i·13-s + (0.530 − 0.635i)15-s + 1.04i·17-s + 1.61·19-s + 1.26·21-s + 0.133i·23-s + (0.178 + 0.983i)25-s − 1.08i·27-s + 1.43·29-s − 0.398·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.640 - 0.767i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ 0.640 - 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56754 + 0.733345i\)
\(L(\frac12)\) \(\approx\) \(1.56754 + 0.733345i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-8.58 - 7.16i)T \)
good3 \( 1 + 4.30iT - 27T^{2} \)
7 \( 1 - 28.3iT - 343T^{2} \)
11 \( 1 + 65.2T + 1.33e3T^{2} \)
13 \( 1 - 33.6iT - 2.19e3T^{2} \)
17 \( 1 - 73.3iT - 4.91e3T^{2} \)
19 \( 1 - 134.T + 6.85e3T^{2} \)
23 \( 1 - 14.7iT - 1.21e4T^{2} \)
29 \( 1 - 224.T + 2.43e4T^{2} \)
31 \( 1 + 68.8T + 2.97e4T^{2} \)
37 \( 1 + 196. iT - 5.06e4T^{2} \)
41 \( 1 + 143.T + 6.89e4T^{2} \)
43 \( 1 + 15.0iT - 7.95e4T^{2} \)
47 \( 1 - 134. iT - 1.03e5T^{2} \)
53 \( 1 + 262. iT - 1.48e5T^{2} \)
59 \( 1 + 119.T + 2.05e5T^{2} \)
61 \( 1 - 16.5T + 2.26e5T^{2} \)
67 \( 1 - 545. iT - 3.00e5T^{2} \)
71 \( 1 - 199.T + 3.57e5T^{2} \)
73 \( 1 + 43.2iT - 3.89e5T^{2} \)
79 \( 1 - 438.T + 4.93e5T^{2} \)
83 \( 1 + 1.22e3iT - 5.71e5T^{2} \)
89 \( 1 + 723.T + 7.04e5T^{2} \)
97 \( 1 + 1.13e3iT - 9.12e5T^{2} \)
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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

−12.65913544294024325006909857592, −11.76396102502967852505961173485, −10.50323286512021716045967702258, −9.563089885061729664199951097645, −8.317864945652795789230397665491, −7.25424642278877181620480613383, −6.09789974954694378275072507862, −5.23731802382952637500698176788, −2.84104206802738002734833689816, −1.86548085766597083425000003769, 0.847848186583845727473206985292, 3.08165116074695062012308450986, 4.68967609455711988711723475681, 5.30427269549830813412618538750, 7.12550699770120201755888472927, 8.090131359893628820628692306781, 9.664969038565207790114155424853, 10.13865607151915964124497734979, 10.86542309279719579078972187617, 12.46854833806434664708545142805

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