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Properties

Label 2-160-160.109-c1-0-8
Degree $2$
Conductor $160$
Sign $0.863 + 0.504i$
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  + (−0.125 − 1.40i)2-s + (1.86 + 0.772i)3-s + (−1.96 + 0.353i)4-s + (1.59 + 1.56i)5-s + (0.854 − 2.72i)6-s + (0.798 + 0.798i)7-s + (0.745 + 2.72i)8-s + (0.760 + 0.760i)9-s + (2.00 − 2.44i)10-s + (−1.21 − 2.92i)11-s + (−3.94 − 0.861i)12-s + (0.175 − 0.424i)13-s + (1.02 − 1.22i)14-s + (1.76 + 4.15i)15-s + (3.74 − 1.39i)16-s − 3.66·17-s + ⋯
L(s)  = 1  + (−0.0887 − 0.996i)2-s + (1.07 + 0.446i)3-s + (−0.984 + 0.176i)4-s + (0.713 + 0.701i)5-s + (0.348 − 1.11i)6-s + (0.301 + 0.301i)7-s + (0.263 + 0.964i)8-s + (0.253 + 0.253i)9-s + (0.635 − 0.772i)10-s + (−0.365 − 0.882i)11-s + (−1.13 − 0.248i)12-s + (0.0487 − 0.117i)13-s + (0.273 − 0.327i)14-s + (0.455 + 1.07i)15-s + (0.937 − 0.348i)16-s − 0.889·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.863 + 0.504i$
Analytic conductor: \(1.27760\)
Root analytic conductor: \(1.13031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1/2),\ 0.863 + 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39169 - 0.376859i\)
\(L(\frac12)\) \(\approx\) \(1.39169 - 0.376859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.125 + 1.40i)T \)
5 \( 1 + (-1.59 - 1.56i)T \)
good3 \( 1 + (-1.86 - 0.772i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.798 - 0.798i)T + 7iT^{2} \)
11 \( 1 + (1.21 + 2.92i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.175 + 0.424i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 + (5.39 + 2.23i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.04 + 3.04i)T - 23iT^{2} \)
29 \( 1 + (-0.995 + 2.40i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 2.13T + 31T^{2} \)
37 \( 1 + (-4.33 - 10.4i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.79 - 6.79i)T + 41iT^{2} \)
43 \( 1 + (9.19 - 3.80i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 2.81T + 47T^{2} \)
53 \( 1 + (-0.347 + 0.144i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-10.0 + 4.16i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (5.16 - 12.4i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (7.97 + 3.30i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-5.88 + 5.88i)T - 71iT^{2} \)
73 \( 1 + (-4.50 + 4.50i)T - 73iT^{2} \)
79 \( 1 + 5.05iT - 79T^{2} \)
83 \( 1 + (2.81 - 6.80i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.44 + 7.44i)T - 89iT^{2} \)
97 \( 1 - 1.85iT - 97T^{2} \)
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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

−13.18635406244725917463618013178, −11.51110705592938470504323169342, −10.74840344459437841632280674371, −9.807317051064940281482802146172, −8.845996445671147277284531388977, −8.206366291841056423451163945214, −6.28539524141630842240993010670, −4.66180074653050036506346129983, −3.16913802097703319970738353192, −2.33848225015655733362578279359, 1.98503680561912132319906556922, 4.21091696832317573024328320661, 5.42573347802909362254918139143, 6.84637646712302937471094000772, 7.82934848114679945780574882210, 8.746218949357727639539216946519, 9.406499945009954057214027460178, 10.65252069314370456623696886809, 12.71807952505341745465992568731, 13.10375152937265706034994366384

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