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Properties

Label 2-160-40.13-c2-0-2
Degree $2$
Conductor $160$
Sign $-0.332 - 0.943i$
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  + (−0.130 + 0.130i)3-s + (−4.38 + 2.39i)5-s + (1.59 − 1.59i)7-s + 8.96i·9-s + 11.9i·11-s + (−9.59 + 9.59i)13-s + (0.260 − 0.887i)15-s + (0.857 − 0.857i)17-s − 20.5·19-s + 0.417i·21-s + (22.1 + 22.1i)23-s + (13.4 − 21.0i)25-s + (−2.34 − 2.34i)27-s + 27.3·29-s − 40.0·31-s + ⋯
L(s)  = 1  + (−0.0435 + 0.0435i)3-s + (−0.877 + 0.479i)5-s + (0.228 − 0.228i)7-s + 0.996i·9-s + 1.08i·11-s + (−0.738 + 0.738i)13-s + (0.0173 − 0.0591i)15-s + (0.0504 − 0.0504i)17-s − 1.08·19-s + 0.0198i·21-s + (0.963 + 0.963i)23-s + (0.539 − 0.841i)25-s + (−0.0870 − 0.0870i)27-s + 0.942·29-s − 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.332 - 0.943i$
Analytic conductor: \(4.35968\)
Root analytic conductor: \(2.08798\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1),\ -0.332 - 0.943i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.539917 + 0.762793i\)
\(L(\frac12)\) \(\approx\) \(0.539917 + 0.762793i\)
\(L(2)\) 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 \( 1 + (4.38 - 2.39i)T \)
good3 \( 1 + (0.130 - 0.130i)T - 9iT^{2} \)
7 \( 1 + (-1.59 + 1.59i)T - 49iT^{2} \)
11 \( 1 - 11.9iT - 121T^{2} \)
13 \( 1 + (9.59 - 9.59i)T - 169iT^{2} \)
17 \( 1 + (-0.857 + 0.857i)T - 289iT^{2} \)
19 \( 1 + 20.5T + 361T^{2} \)
23 \( 1 + (-22.1 - 22.1i)T + 529iT^{2} \)
29 \( 1 - 27.3T + 841T^{2} \)
31 \( 1 + 40.0T + 961T^{2} \)
37 \( 1 + (-1.57 - 1.57i)T + 1.36e3iT^{2} \)
41 \( 1 + 37.5T + 1.68e3T^{2} \)
43 \( 1 + (-49.2 + 49.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-34.0 + 34.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (-28.8 + 28.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 92.7T + 3.48e3T^{2} \)
61 \( 1 - 4.82iT - 3.72e3T^{2} \)
67 \( 1 + (54.6 + 54.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 59.2T + 5.04e3T^{2} \)
73 \( 1 + (-34.1 - 34.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 96.2iT - 6.24e3T^{2} \)
83 \( 1 + (63.6 - 63.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 3.68iT - 7.92e3T^{2} \)
97 \( 1 + (-46.0 + 46.0i)T - 9.40e3iT^{2} \)
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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

−12.84369463834724353663668494558, −11.89374987722834512036115980976, −10.96064038006591122060749533554, −10.13980649073430894108779930377, −8.776297222349283889255150431266, −7.51411218384575551749050042423, −6.97414865053259331500352339959, −5.06579684524839429467756008804, −4.06906897173417865641900238928, −2.24651062898452295193772436094, 0.58762804741124329717946617472, 3.09579189414297609792862456427, 4.44729683334388917431073970581, 5.79528813756298342758545654047, 7.10767735189381235548712387980, 8.373061297091634734275981019421, 8.986974582799736243006804961035, 10.49105458622746309430492287240, 11.45794239954977175789751683833, 12.39057198813948275116837413627

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