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Properties

Label 2-80-5.4-c19-0-47
Degree $2$
Conductor $80$
Sign $-0.990 + 0.137i$
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  − 6.40e4i·3-s + (4.32e6 − 5.99e5i)5-s − 1.31e6i·7-s − 2.93e9·9-s + 1.19e10·11-s − 3.94e10i·13-s + (−3.83e10 − 2.77e11i)15-s + 8.42e9i·17-s + 7.32e11·19-s − 8.40e10·21-s − 5.55e12i·23-s + (1.83e13 − 5.18e12i)25-s + 1.13e14i·27-s + 4.72e13·29-s + 2.08e14·31-s + ⋯
L(s)  = 1  − 1.87i·3-s + (0.990 − 0.137i)5-s − 0.0122i·7-s − 2.52·9-s + 1.52·11-s − 1.03i·13-s + (−0.257 − 1.86i)15-s + 0.0172i·17-s + 0.520·19-s − 0.0231·21-s − 0.643i·23-s + (0.962 − 0.271i)25-s + 2.87i·27-s + 0.605·29-s + 1.41·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.990 + 0.137i$
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: \((2,\ 80,\ (\ :19/2),\ -0.990 + 0.137i)\)

Particular Values

\(L(10)\) \(\approx\) \(3.140495907\)
\(L(\frac12)\) \(\approx\) \(3.140495907\)
\(L(\frac{21}{2})\) 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 \( 1 + (-4.32e6 + 5.99e5i)T \)
good3 \( 1 + 6.40e4iT - 1.16e9T^{2} \)
7 \( 1 + 1.31e6iT - 1.13e16T^{2} \)
11 \( 1 - 1.19e10T + 6.11e19T^{2} \)
13 \( 1 + 3.94e10iT - 1.46e21T^{2} \)
17 \( 1 - 8.42e9iT - 2.39e23T^{2} \)
19 \( 1 - 7.32e11T + 1.97e24T^{2} \)
23 \( 1 + 5.55e12iT - 7.46e25T^{2} \)
29 \( 1 - 4.72e13T + 6.10e27T^{2} \)
31 \( 1 - 2.08e14T + 2.16e28T^{2} \)
37 \( 1 - 1.14e15iT - 6.24e29T^{2} \)
41 \( 1 + 2.76e15T + 4.39e30T^{2} \)
43 \( 1 + 5.39e15iT - 1.08e31T^{2} \)
47 \( 1 + 6.96e14iT - 5.88e31T^{2} \)
53 \( 1 + 2.06e16iT - 5.77e32T^{2} \)
59 \( 1 - 2.26e16T + 4.42e33T^{2} \)
61 \( 1 + 1.05e16T + 8.34e33T^{2} \)
67 \( 1 + 2.23e17iT - 4.95e34T^{2} \)
71 \( 1 - 6.99e16T + 1.49e35T^{2} \)
73 \( 1 + 7.86e17iT - 2.53e35T^{2} \)
79 \( 1 - 1.65e18T + 1.13e36T^{2} \)
83 \( 1 + 1.02e18iT - 2.90e36T^{2} \)
89 \( 1 + 2.27e18T + 1.09e37T^{2} \)
97 \( 1 - 4.85e18iT - 5.60e37T^{2} \)
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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

−10.29327055383109671242927874909, −8.926140350721417784404683658806, −8.088273486644143000452290569411, −6.75425688271776177481580587585, −6.32574303780544164297436361357, −5.18587764751281591321878784161, −3.17453730378773192605081718683, −2.13781063224957021621012310899, −1.21952242999787082750083353528, −0.62463987325054914952461666930, 1.22550760096015955110038562421, 2.64379808852819843500230102728, 3.78029199236255698988096105613, 4.59261214945095700665383941763, 5.67710460912577425808711266190, 6.64910095776490756395902719946, 8.694588604873487340335673547936, 9.431962750604496384352990064072, 9.956992611845700088702194684213, 11.11191299827170541989772468616

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