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Properties

Label 2-80-80.69-c1-0-5
Degree $2$
Conductor $80$
Sign $0.880 + 0.474i$
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  + (0.490 − 1.32i)2-s + (1.99 + 1.99i)3-s + (−1.51 − 1.30i)4-s + (−2.16 − 0.569i)5-s + (3.61 − 1.66i)6-s + 1.09·7-s + (−2.46 + 1.37i)8-s + 4.93i·9-s + (−1.81 + 2.58i)10-s + (−2.33 − 2.33i)11-s + (−0.437 − 5.61i)12-s + (−1.80 − 1.80i)13-s + (0.534 − 1.44i)14-s + (−3.17 − 5.44i)15-s + (0.619 + 3.95i)16-s + 4.93i·17-s + ⋯
L(s)  = 1  + (0.346 − 0.938i)2-s + (1.14 + 1.14i)3-s + (−0.759 − 0.650i)4-s + (−0.966 − 0.254i)5-s + (1.47 − 0.680i)6-s + 0.412·7-s + (−0.873 + 0.487i)8-s + 1.64i·9-s + (−0.574 + 0.818i)10-s + (−0.703 − 0.703i)11-s + (−0.126 − 1.62i)12-s + (−0.501 − 0.501i)13-s + (0.142 − 0.386i)14-s + (−0.818 − 1.40i)15-s + (0.154 + 0.987i)16-s + 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.880 + 0.474i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.880 + 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19607 - 0.302138i\)
\(L(\frac12)\) \(\approx\) \(1.19607 - 0.302138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.490 + 1.32i)T \)
5 \( 1 + (2.16 + 0.569i)T \)
good3 \( 1 + (-1.99 - 1.99i)T + 3iT^{2} \)
7 \( 1 - 1.09T + 7T^{2} \)
11 \( 1 + (2.33 + 2.33i)T + 11iT^{2} \)
13 \( 1 + (1.80 + 1.80i)T + 13iT^{2} \)
17 \( 1 - 4.93iT - 17T^{2} \)
19 \( 1 + (-2.03 + 2.03i)T - 19iT^{2} \)
23 \( 1 + 1.45T + 23T^{2} \)
29 \( 1 + (0.707 - 0.707i)T - 29iT^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + (4.35 - 4.35i)T - 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-2.22 + 2.22i)T - 43iT^{2} \)
47 \( 1 + 2.09iT - 47T^{2} \)
53 \( 1 + (0.215 - 0.215i)T - 53iT^{2} \)
59 \( 1 + (1.16 + 1.16i)T + 59iT^{2} \)
61 \( 1 + (3.46 - 3.46i)T - 61iT^{2} \)
67 \( 1 + (-5.04 - 5.04i)T + 67iT^{2} \)
71 \( 1 - 6.40iT - 71T^{2} \)
73 \( 1 + 5.24T + 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 + (5.67 + 5.67i)T + 83iT^{2} \)
89 \( 1 - 6.87iT - 89T^{2} \)
97 \( 1 + 3.77iT - 97T^{2} \)
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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

−14.32884406977739065431452367554, −13.34308007879093733475235366202, −12.07724161997892262102972669061, −10.84758415589388918929266661413, −10.08210207349366531212762223384, −8.716468017906150011604131822164, −8.087125105082376760266939909266, −5.14916230250929357164987636593, −4.03081793542534553811673865677, −2.92195415866566169542857055225, 2.90833175246742863955935257584, 4.65206840014813268123618430060, 6.76949737527615949497877825070, 7.64242927566893300050131230242, 8.159677949191236158200602711790, 9.538337964070882368457714740061, 11.79697692591194789312303722776, 12.57071896219854232457690576546, 13.72639447222628438449124209202, 14.40597191930621962980458603190

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