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Properties

Label 2-80-16.5-c1-0-1
Degree $2$
Conductor $80$
Sign $0.259 - 0.965i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
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  + (−1.37 + 0.320i)2-s + (−0.720 + 0.720i)3-s + (1.79 − 0.883i)4-s + (0.707 + 0.707i)5-s + (0.761 − 1.22i)6-s + 4.02i·7-s + (−2.18 + 1.79i)8-s + 1.96i·9-s + (−1.20 − 0.747i)10-s + (−0.646 − 0.646i)11-s + (−0.656 + 1.92i)12-s + (4.91 − 4.91i)13-s + (−1.29 − 5.54i)14-s − 1.01·15-s + (2.43 − 3.17i)16-s − 2.70·17-s + ⋯
L(s)  = 1  + (−0.973 + 0.226i)2-s + (−0.416 + 0.416i)3-s + (0.897 − 0.441i)4-s + (0.316 + 0.316i)5-s + (0.310 − 0.499i)6-s + 1.52i·7-s + (−0.773 + 0.633i)8-s + 0.653i·9-s + (−0.379 − 0.236i)10-s + (−0.195 − 0.195i)11-s + (−0.189 + 0.557i)12-s + (1.36 − 1.36i)13-s + (−0.345 − 1.48i)14-s − 0.263·15-s + (0.609 − 0.792i)16-s − 0.656·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.259 - 0.965i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474312 + 0.363579i\)
\(L(\frac12)\) \(\approx\) \(0.474312 + 0.363579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.320i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.720 - 0.720i)T - 3iT^{2} \)
7 \( 1 - 4.02iT - 7T^{2} \)
11 \( 1 + (0.646 + 0.646i)T + 11iT^{2} \)
13 \( 1 + (-4.91 + 4.91i)T - 13iT^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + (0.438 - 0.438i)T - 19iT^{2} \)
23 \( 1 + 3.60iT - 23T^{2} \)
29 \( 1 + (-2.00 + 2.00i)T - 29iT^{2} \)
31 \( 1 - 4.30T + 31T^{2} \)
37 \( 1 + (0.743 + 0.743i)T + 37iT^{2} \)
41 \( 1 - 0.603iT - 41T^{2} \)
43 \( 1 + (5.03 + 5.03i)T + 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (-4.07 - 4.07i)T + 53iT^{2} \)
59 \( 1 + (-1.22 - 1.22i)T + 59iT^{2} \)
61 \( 1 + (6.98 - 6.98i)T - 61iT^{2} \)
67 \( 1 + (-5.24 + 5.24i)T - 67iT^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 - 1.30iT - 73T^{2} \)
79 \( 1 - 0.611T + 79T^{2} \)
83 \( 1 + (-1.29 + 1.29i)T - 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−15.31086868097340381268975576101, −13.60374074808582237255155971363, −12.14305616995706872113181182057, −10.95289056152024646132392210821, −10.33107372396402994937059379504, −8.903093601010574580808893302832, −8.101286405768503335422096350196, −6.22957795488557128833993290771, −5.45035496836055385583681898823, −2.56040380100212905166033433798, 1.27794611150868417872909509578, 3.94498258765149888742711374704, 6.37985419476813256100835789574, 7.12702016261729048507604825085, 8.619824817587449666388945297995, 9.721073725190053844788809779511, 10.89132235413497354020025377869, 11.69984601475331558247888847965, 13.01570396244850904397974789164, 13.92241921590225480842768759799

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