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Properties

Label 2-80-5.4-c19-0-16
Degree $2$
Conductor $80$
Sign $0.128 - 0.991i$
Analytic cond. $183.053$
Root an. cond. $13.5297$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.17e4i·3-s + (5.60e5 − 4.33e6i)5-s − 4.95e7i·7-s − 2.64e9·9-s + 3.28e9·11-s + 6.48e9i·13-s + (2.67e11 + 3.45e10i)15-s + 2.26e11i·17-s − 2.35e12·19-s + 3.05e12·21-s − 1.37e13i·23-s + (−1.84e13 − 4.85e12i)25-s − 9.14e13i·27-s − 8.06e12·29-s − 8.98e13·31-s + ⋯
L(s)  = 1  + 1.80i·3-s + (0.128 − 0.991i)5-s − 0.463i·7-s − 2.27·9-s + 0.419·11-s + 0.169i·13-s + (1.79 + 0.232i)15-s + 0.462i·17-s − 1.67·19-s + 0.839·21-s − 1.59i·23-s + (−0.967 − 0.254i)25-s − 2.30i·27-s − 0.103·29-s − 0.610·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.128 - 0.991i$
Analytic conductor: \(183.053\)
Root analytic conductor: \(13.5297\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :19/2),\ 0.128 - 0.991i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.570056826\)
\(L(\frac12)\) \(\approx\) \(1.570056826\)
\(L(\frac{21}{2})\) 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 \( 1 + (-5.60e5 + 4.33e6i)T \)
good3 \( 1 - 6.17e4iT - 1.16e9T^{2} \)
7 \( 1 + 4.95e7iT - 1.13e16T^{2} \)
11 \( 1 - 3.28e9T + 6.11e19T^{2} \)
13 \( 1 - 6.48e9iT - 1.46e21T^{2} \)
17 \( 1 - 2.26e11iT - 2.39e23T^{2} \)
19 \( 1 + 2.35e12T + 1.97e24T^{2} \)
23 \( 1 + 1.37e13iT - 7.46e25T^{2} \)
29 \( 1 + 8.06e12T + 6.10e27T^{2} \)
31 \( 1 + 8.98e13T + 2.16e28T^{2} \)
37 \( 1 - 1.07e15iT - 6.24e29T^{2} \)
41 \( 1 + 1.28e15T + 4.39e30T^{2} \)
43 \( 1 + 4.58e15iT - 1.08e31T^{2} \)
47 \( 1 - 2.75e15iT - 5.88e31T^{2} \)
53 \( 1 - 1.29e15iT - 5.77e32T^{2} \)
59 \( 1 - 1.16e17T + 4.42e33T^{2} \)
61 \( 1 + 7.11e15T + 8.34e33T^{2} \)
67 \( 1 - 3.53e17iT - 4.95e34T^{2} \)
71 \( 1 - 4.49e17T + 1.49e35T^{2} \)
73 \( 1 - 4.92e17iT - 2.53e35T^{2} \)
79 \( 1 - 9.26e17T + 1.13e36T^{2} \)
83 \( 1 + 3.08e18iT - 2.90e36T^{2} \)
89 \( 1 - 5.14e18T + 1.09e37T^{2} \)
97 \( 1 - 7.93e18iT - 5.60e37T^{2} \)
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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

−10.67149846882774240847699297230, −10.07864443001399545988682141769, −8.903115342862530509092993279427, −8.435664125511641812664878650622, −6.41600374192483134983727257410, −5.21033898730009000013405475792, −4.32577921417715566264641374893, −3.80168979553658751836982607076, −2.21527407014157450515786027568, −0.61316444074199639590785436108, 0.44038540943630401582193740353, 1.77258717714417326844245566405, 2.34858118634744293245888348181, 3.51987012624285432707908388996, 5.58473086958657196268320308049, 6.43231031215566835038137541143, 7.18960053431098431196230476577, 8.090107996725440910204401240320, 9.287563532024872407021091452807, 10.90673362443933978273081459676

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