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Properties

Label 2-80-5.3-c16-0-32
Degree $2$
Conductor $80$
Sign $0.869 - 0.494i$
Analytic cond. $129.859$
Root an. cond. $11.3955$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.46e3 + 5.46e3i)3-s + (1.87e5 − 3.42e5i)5-s + (−6.16e6 + 6.16e6i)7-s + 1.66e7i·9-s + 3.83e8·11-s + (6.97e8 + 6.97e8i)13-s + (2.89e9 − 8.49e8i)15-s + (2.15e9 − 2.15e9i)17-s − 2.01e10i·19-s − 6.73e10·21-s + (3.86e10 + 3.86e10i)23-s + (−8.23e10 − 1.28e11i)25-s + (1.44e11 − 1.44e11i)27-s − 2.47e11i·29-s − 1.30e12·31-s + ⋯
L(s)  = 1  + (0.832 + 0.832i)3-s + (0.479 − 0.877i)5-s + (−1.06 + 1.06i)7-s + 0.387i·9-s + 1.78·11-s + (0.855 + 0.855i)13-s + (1.13 − 0.331i)15-s + (0.309 − 0.309i)17-s − 1.18i·19-s − 1.78·21-s + (0.493 + 0.493i)23-s + (−0.539 − 0.841i)25-s + (0.510 − 0.510i)27-s − 0.494i·29-s − 1.52·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(129.859\)
Root analytic conductor: \(11.3955\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :8),\ 0.869 - 0.494i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(3.748037399\)
\(L(\frac12)\) \(\approx\) \(3.748037399\)
\(L(9)\) 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 \( 1 + (-1.87e5 + 3.42e5i)T \)
good3 \( 1 + (-5.46e3 - 5.46e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (6.16e6 - 6.16e6i)T - 3.32e13iT^{2} \)
11 \( 1 - 3.83e8T + 4.59e16T^{2} \)
13 \( 1 + (-6.97e8 - 6.97e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-2.15e9 + 2.15e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 2.01e10iT - 2.88e20T^{2} \)
23 \( 1 + (-3.86e10 - 3.86e10i)T + 6.13e21iT^{2} \)
29 \( 1 + 2.47e11iT - 2.50e23T^{2} \)
31 \( 1 + 1.30e12T + 7.27e23T^{2} \)
37 \( 1 + (-2.17e12 + 2.17e12i)T - 1.23e25iT^{2} \)
41 \( 1 - 1.11e12T + 6.37e25T^{2} \)
43 \( 1 + (-9.81e11 - 9.81e11i)T + 1.36e26iT^{2} \)
47 \( 1 + (-9.89e12 + 9.89e12i)T - 5.66e26iT^{2} \)
53 \( 1 + (-1.15e13 - 1.15e13i)T + 3.87e27iT^{2} \)
59 \( 1 - 1.15e14iT - 2.15e28T^{2} \)
61 \( 1 - 1.18e12T + 3.67e28T^{2} \)
67 \( 1 + (-6.21e13 + 6.21e13i)T - 1.64e29iT^{2} \)
71 \( 1 - 7.46e14T + 4.16e29T^{2} \)
73 \( 1 + (1.01e14 + 1.01e14i)T + 6.50e29iT^{2} \)
79 \( 1 - 3.00e15iT - 2.30e30T^{2} \)
83 \( 1 + (-5.42e13 - 5.42e13i)T + 5.07e30iT^{2} \)
89 \( 1 + 1.91e14iT - 1.54e31T^{2} \)
97 \( 1 + (-5.22e15 + 5.22e15i)T - 6.14e31iT^{2} \)
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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

−11.41968234334896406397010061031, −9.572174543269482062023460678663, −9.187786445262142808691526619183, −8.786095380284905563899794713650, −6.71358630039507905773907021669, −5.72215602094889235268616620996, −4.27796206352156636730446308921, −3.43749240194790950076714937234, −2.18281180427280630870618806024, −0.874665353010586222246963394043, 0.910868617636270672431199865465, 1.76145206478319502931353799647, 3.24749710850687258501248259248, 3.70996837384959741240127545794, 6.06674327028023916124346504594, 6.77910082102033036284703526576, 7.65756813860269096847966835754, 8.964958410119685360623936172280, 10.05196147704909861089446357577, 10.97718392965748383846938962436

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