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Properties

Label 2-160-160.109-c1-0-10
Degree $2$
Conductor $160$
Sign $0.760 - 0.648i$
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  + (1.36 + 0.377i)2-s + (0.518 + 0.214i)3-s + (1.71 + 1.03i)4-s + (−1.73 + 1.41i)5-s + (0.625 + 0.488i)6-s + (0.589 + 0.589i)7-s + (1.94 + 2.05i)8-s + (−1.89 − 1.89i)9-s + (−2.89 + 1.27i)10-s + (−1.01 − 2.44i)11-s + (0.667 + 0.901i)12-s + (1.71 − 4.13i)13-s + (0.580 + 1.02i)14-s + (−1.20 + 0.360i)15-s + (1.87 + 3.53i)16-s + 6.07·17-s + ⋯
L(s)  = 1  + (0.963 + 0.267i)2-s + (0.299 + 0.123i)3-s + (0.857 + 0.515i)4-s + (−0.774 + 0.632i)5-s + (0.255 + 0.199i)6-s + (0.222 + 0.222i)7-s + (0.688 + 0.725i)8-s + (−0.632 − 0.632i)9-s + (−0.915 + 0.402i)10-s + (−0.305 − 0.738i)11-s + (0.192 + 0.260i)12-s + (0.474 − 1.14i)13-s + (0.155 + 0.274i)14-s + (−0.310 + 0.0931i)15-s + (0.469 + 0.882i)16-s + 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.760 - 0.648i$
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: \((2,\ 160,\ (\ :1/2),\ 0.760 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75091 + 0.645347i\)
\(L(\frac12)\) \(\approx\) \(1.75091 + 0.645347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.377i)T \)
5 \( 1 + (1.73 - 1.41i)T \)
good3 \( 1 + (-0.518 - 0.214i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.589 - 0.589i)T + 7iT^{2} \)
11 \( 1 + (1.01 + 2.44i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.71 + 4.13i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.07T + 17T^{2} \)
19 \( 1 + (6.06 + 2.51i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.31 - 5.31i)T - 23iT^{2} \)
29 \( 1 + (2.26 - 5.46i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 3.85T + 31T^{2} \)
37 \( 1 + (-0.669 - 1.61i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.86 - 1.86i)T + 41iT^{2} \)
43 \( 1 + (8.18 - 3.39i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 0.751T + 47T^{2} \)
53 \( 1 + (-0.406 + 0.168i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.14 - 1.71i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.00834 - 0.0201i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-11.6 - 4.80i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (4.12 - 4.12i)T - 71iT^{2} \)
73 \( 1 + (-4.17 + 4.17i)T - 73iT^{2} \)
79 \( 1 - 1.62iT - 79T^{2} \)
83 \( 1 + (-0.606 + 1.46i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-3.01 + 3.01i)T - 89iT^{2} \)
97 \( 1 - 4.53iT - 97T^{2} \)
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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

−13.07559989255429807129437251019, −12.01715370209579181680519698189, −11.28556703128943237684355006963, −10.30733738809399834782992726038, −8.412086577549863003384916687689, −7.85112492636294755179694987132, −6.40044547996307633587110835568, −5.44034042061400446112690273801, −3.71616031187280934379743447710, −2.97896419289952519286028397065, 2.04584476993835077362570308207, 3.86121593253635479217774052703, 4.76342400761128013886360139840, 6.13317838060462576446294571627, 7.58798578543155551448564901028, 8.398343628565217152962830348470, 9.992143101387160704541993346099, 11.06871865977422435319917566295, 12.02433681725875683425846516590, 12.66051652311661165483025808064

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