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Properties

Label 2-160-160.107-c1-0-13
Degree $2$
Conductor $160$
Sign $0.379 + 0.925i$
Analytic cond. $1.27760$
Root an. cond. $1.13031$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0556 − 1.41i)2-s + (0.532 + 0.220i)3-s + (−1.99 − 0.157i)4-s + (2.20 + 0.363i)5-s + (0.341 − 0.740i)6-s + 3.48·7-s + (−0.333 + 2.80i)8-s + (−1.88 − 1.88i)9-s + (0.637 − 3.09i)10-s + (0.482 − 1.16i)11-s + (−1.02 − 0.523i)12-s + (−5.37 − 2.22i)13-s + (0.193 − 4.92i)14-s + (1.09 + 0.680i)15-s + (3.95 + 0.627i)16-s + (−1.35 + 1.35i)17-s + ⋯
L(s)  = 1  + (0.0393 − 0.999i)2-s + (0.307 + 0.127i)3-s + (−0.996 − 0.0786i)4-s + (0.986 + 0.162i)5-s + (0.139 − 0.302i)6-s + 1.31·7-s + (−0.117 + 0.993i)8-s + (−0.628 − 0.628i)9-s + (0.201 − 0.979i)10-s + (0.145 − 0.351i)11-s + (−0.296 − 0.151i)12-s + (−1.49 − 0.617i)13-s + (0.0518 − 1.31i)14-s + (0.282 + 0.175i)15-s + (0.987 + 0.156i)16-s + (−0.329 + 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.379 + 0.925i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11712 - 0.748954i\)
\(L(\frac12)\) \(\approx\) \(1.11712 - 0.748954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0556 + 1.41i)T \)
5 \( 1 + (-2.20 - 0.363i)T \)
good3 \( 1 + (-0.532 - 0.220i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 - 3.48T + 7T^{2} \)
11 \( 1 + (-0.482 + 1.16i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (5.37 + 2.22i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (1.35 - 1.35i)T - 17iT^{2} \)
19 \( 1 + (-2.64 - 6.39i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + 1.76T + 23T^{2} \)
29 \( 1 + (-0.00445 - 0.0107i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 2.36iT - 31T^{2} \)
37 \( 1 + (10.3 - 4.28i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.39 + 3.39i)T - 41iT^{2} \)
43 \( 1 + (-3.45 - 8.32i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (4.73 + 4.73i)T + 47iT^{2} \)
53 \( 1 + (4.21 - 1.74i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (1.69 - 4.08i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (3.79 - 1.57i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.83 - 6.84i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-10.0 + 10.0i)T - 71iT^{2} \)
73 \( 1 + 8.10iT - 73T^{2} \)
79 \( 1 + 0.602iT - 79T^{2} \)
83 \( 1 + (-4.47 + 10.7i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.30 + 7.30i)T - 89iT^{2} \)
97 \( 1 + (-0.614 - 0.614i)T + 97iT^{2} \)
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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

−12.47613290111232279299677961085, −11.76660620033047480104013945327, −10.61613793836561213010093995253, −9.846630743344656953844496142261, −8.826606334013711638377276928600, −7.85368609787005789134098602606, −5.84487241663126532917314805780, −4.85878162578607358257747930305, −3.18730008142528509258064038069, −1.80143734039196394383002691584, 2.23712031597276312133530231822, 4.77771319985509766608847426154, 5.26937451409124096531938935578, 6.87604361003367474742004910063, 7.77782016024444396420795784646, 8.900679005710358568752068319883, 9.626457718056860060762094179120, 11.03949666635034364421559320319, 12.30420829890151020567908930034, 13.55520128118291149657517743109

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