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Properties

Label 2-160-5.2-c2-0-2
Degree $2$
Conductor $160$
Sign $-0.267 - 0.963i$
Analytic cond. $4.35968$
Root an. cond. $2.08798$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.89 + 2.89i)3-s + (4.38 − 2.40i)5-s + (5.86 + 5.86i)7-s − 7.76i·9-s − 7.84·11-s + (−15.5 + 15.5i)13-s + (−5.71 + 19.6i)15-s + (13.3 + 13.3i)17-s + 21.1i·19-s − 33.9·21-s + (2.28 − 2.28i)23-s + (13.3 − 21.1i)25-s + (−3.58 − 3.58i)27-s − 7.68i·29-s − 32.1·31-s + ⋯
L(s)  = 1  + (−0.965 + 0.965i)3-s + (0.876 − 0.481i)5-s + (0.838 + 0.838i)7-s − 0.862i·9-s − 0.713·11-s + (−1.19 + 1.19i)13-s + (−0.380 + 1.31i)15-s + (0.788 + 0.788i)17-s + 1.11i·19-s − 1.61·21-s + (0.0994 − 0.0994i)23-s + (0.535 − 0.844i)25-s + (−0.132 − 0.132i)27-s − 0.265i·29-s − 1.03·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.267 - 0.963i$
Analytic conductor: \(4.35968\)
Root analytic conductor: \(2.08798\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1),\ -0.267 - 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.683060 + 0.898513i\)
\(L(\frac12)\) \(\approx\) \(0.683060 + 0.898513i\)
\(L(2)\) 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 \( 1 + (-4.38 + 2.40i)T \)
good3 \( 1 + (2.89 - 2.89i)T - 9iT^{2} \)
7 \( 1 + (-5.86 - 5.86i)T + 49iT^{2} \)
11 \( 1 + 7.84T + 121T^{2} \)
13 \( 1 + (15.5 - 15.5i)T - 169iT^{2} \)
17 \( 1 + (-13.3 - 13.3i)T + 289iT^{2} \)
19 \( 1 - 21.1iT - 361T^{2} \)
23 \( 1 + (-2.28 + 2.28i)T - 529iT^{2} \)
29 \( 1 + 7.68iT - 841T^{2} \)
31 \( 1 + 32.1T + 961T^{2} \)
37 \( 1 + (-38.2 - 38.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 20.8T + 1.68e3T^{2} \)
43 \( 1 + (-57.9 + 57.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-35.1 - 35.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (22.8 - 22.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 69.7iT - 3.48e3T^{2} \)
61 \( 1 + 11.9T + 3.72e3T^{2} \)
67 \( 1 + (27.5 + 27.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 52.2T + 5.04e3T^{2} \)
73 \( 1 + (4.73 - 4.73i)T - 5.32e3iT^{2} \)
79 \( 1 + 31.3iT - 6.24e3T^{2} \)
83 \( 1 + (-33.3 + 33.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 0.623iT - 7.92e3T^{2} \)
97 \( 1 + (83.1 + 83.1i)T + 9.40e3iT^{2} \)
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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

−12.59734753792924275809892737854, −11.95053758026328169318656406825, −10.85316392424296864820743812615, −9.973339919649911736036788507708, −9.173051013212430442790893466759, −7.85541298920679277661850161076, −5.99514405566107896465305524239, −5.31681546211030341306402650051, −4.42549880176396060971925512909, −2.04993740608906780150189980740, 0.811515661647646139259049879558, 2.58132057925814227102877852892, 5.02031022075198303239577305349, 5.78149850746879665499301074141, 7.29353505140316091171085687184, 7.54319560579074337572299626301, 9.532618881477423114170742990971, 10.65364225864980139630980507933, 11.22228033197422162693265417211, 12.51649832124021939169592465289

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