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Properties

Label 2-80-5.2-c2-0-4
Degree $2$
Conductor $80$
Sign $0.525 + 0.850i$
Analytic cond. $2.17984$
Root an. cond. $1.47642$
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  + (2 − 2i)3-s − 5i·5-s + (−2 − 2i)7-s + i·9-s + 8·11-s + (3 − 3i)13-s + (−10 − 10i)15-s + (7 + 7i)17-s + 20i·19-s − 8·21-s + (2 − 2i)23-s − 25·25-s + (20 + 20i)27-s + 40i·29-s − 52·31-s + ⋯
L(s)  = 1  + (0.666 − 0.666i)3-s − i·5-s + (−0.285 − 0.285i)7-s + 0.111i·9-s + 0.727·11-s + (0.230 − 0.230i)13-s + (−0.666 − 0.666i)15-s + (0.411 + 0.411i)17-s + 1.05i·19-s − 0.380·21-s + (0.0869 − 0.0869i)23-s − 25-s + (0.740 + 0.740i)27-s + 1.37i·29-s − 1.67·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(2.17984\)
Root analytic conductor: \(1.47642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.31940 - 0.735617i\)
\(L(\frac12)\) \(\approx\) \(1.31940 - 0.735617i\)
\(L(2)\) 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 \( 1 + 5iT \)
good3 \( 1 + (-2 + 2i)T - 9iT^{2} \)
7 \( 1 + (2 + 2i)T + 49iT^{2} \)
11 \( 1 - 8T + 121T^{2} \)
13 \( 1 + (-3 + 3i)T - 169iT^{2} \)
17 \( 1 + (-7 - 7i)T + 289iT^{2} \)
19 \( 1 - 20iT - 361T^{2} \)
23 \( 1 + (-2 + 2i)T - 529iT^{2} \)
29 \( 1 - 40iT - 841T^{2} \)
31 \( 1 + 52T + 961T^{2} \)
37 \( 1 + (3 + 3i)T + 1.36e3iT^{2} \)
41 \( 1 + 8T + 1.68e3T^{2} \)
43 \( 1 + (-42 + 42i)T - 1.84e3iT^{2} \)
47 \( 1 + (-18 - 18i)T + 2.20e3iT^{2} \)
53 \( 1 + (-53 + 53i)T - 2.80e3iT^{2} \)
59 \( 1 - 20iT - 3.48e3T^{2} \)
61 \( 1 + 48T + 3.72e3T^{2} \)
67 \( 1 + (62 + 62i)T + 4.48e3iT^{2} \)
71 \( 1 - 28T + 5.04e3T^{2} \)
73 \( 1 + (47 - 47i)T - 5.32e3iT^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + (18 - 18i)T - 6.88e3iT^{2} \)
89 \( 1 - 80iT - 7.92e3T^{2} \)
97 \( 1 + (63 + 63i)T + 9.40e3iT^{2} \)
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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

−13.86044914999809610449259555471, −12.88772438575604322004032726204, −12.21262680069244875304099998371, −10.60743643636833107945835355572, −9.188745117235612226752961263905, −8.295764060792530018351770838178, −7.17546497060046303972154404160, −5.53305507942382811346062691126, −3.74710321636200446913837047237, −1.55983736865652693073481966028, 2.82748992707805876928492998427, 4.06184677211170776568625861228, 6.10199289521861824782559450555, 7.34712988900325271059569299119, 8.983568593412179507645337916561, 9.688183020099762016018134082624, 10.94941113218402555256988984869, 12.00391884968522702685756130805, 13.55539247026847928857388913450, 14.49993930591100624662516488587

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