Properties

Label 2-80-80.67-c1-0-7
Degree $2$
Conductor $80$
Sign $0.999 + 0.0172i$
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  + (1.29 + 0.567i)2-s − 1.96i·3-s + (1.35 + 1.47i)4-s + (−1.72 + 1.42i)5-s + (1.11 − 2.54i)6-s + (−1.60 − 1.60i)7-s + (0.920 + 2.67i)8-s − 0.851·9-s + (−3.04 + 0.861i)10-s + (0.754 + 0.754i)11-s + (2.88 − 2.65i)12-s − 5.94·13-s + (−1.16 − 2.98i)14-s + (2.79 + 3.38i)15-s + (−0.327 + 3.98i)16-s + (1.95 + 1.95i)17-s + ⋯
L(s)  = 1  + (0.915 + 0.401i)2-s − 1.13i·3-s + (0.677 + 0.735i)4-s + (−0.771 + 0.635i)5-s + (0.454 − 1.03i)6-s + (−0.605 − 0.605i)7-s + (0.325 + 0.945i)8-s − 0.283·9-s + (−0.962 + 0.272i)10-s + (0.227 + 0.227i)11-s + (0.833 − 0.767i)12-s − 1.64·13-s + (−0.311 − 0.797i)14-s + (0.720 + 0.874i)15-s + (−0.0817 + 0.996i)16-s + (0.474 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.999 + 0.0172i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.999 + 0.0172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33054 - 0.0114825i\)
\(L(\frac12)\) \(\approx\) \(1.33054 - 0.0114825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.567i)T \)
5 \( 1 + (1.72 - 1.42i)T \)
good3 \( 1 + 1.96iT - 3T^{2} \)
7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
11 \( 1 + (-0.754 - 0.754i)T + 11iT^{2} \)
13 \( 1 + 5.94T + 13T^{2} \)
17 \( 1 + (-1.95 - 1.95i)T + 17iT^{2} \)
19 \( 1 + (0.780 + 0.780i)T + 19iT^{2} \)
23 \( 1 + (-4.93 + 4.93i)T - 23iT^{2} \)
29 \( 1 + (-1.44 + 1.44i)T - 29iT^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 6.93iT - 41T^{2} \)
43 \( 1 + 9.91T + 43T^{2} \)
47 \( 1 + (0.104 - 0.104i)T - 47iT^{2} \)
53 \( 1 - 4.03iT - 53T^{2} \)
59 \( 1 + (-3.46 + 3.46i)T - 59iT^{2} \)
61 \( 1 + (-0.680 - 0.680i)T + 61iT^{2} \)
67 \( 1 + 9.04T + 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + (-2.94 - 2.94i)T + 73iT^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 4.23iT - 83T^{2} \)
89 \( 1 + 0.0426T + 89T^{2} \)
97 \( 1 + (1.91 + 1.91i)T + 97iT^{2} \)
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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

−14.45866973693517102695743219566, −13.17201715935575159402114372178, −12.49705861957749976146240919653, −11.64093526271931617557212504371, −10.21839030918563798514962124811, −8.013839464464968104085620737157, −7.12262333938133328268087075943, −6.54180448429922120307139071759, −4.49245529226561636559035190582, −2.82140471547993004938655216879, 3.17617200714762836863130478872, 4.49710298245524283796164068260, 5.43180204076010283375178432610, 7.29969730758879952722163426020, 9.244801112962775889072524209837, 9.972302160476535361964374714268, 11.37689190116861128257511025657, 12.20772275120229244425354209820, 13.13241862992768029826705079248, 14.73701316422663216858843495782

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