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Properties

Label 2-80-5.4-c19-0-33
Degree $2$
Conductor $80$
Sign $-0.984 + 0.177i$
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  − 6.27e4i·3-s + (−4.29e6 + 7.75e5i)5-s + 1.11e8i·7-s − 2.77e9·9-s − 9.81e9·11-s − 4.48e10i·13-s + (4.86e10 + 2.69e11i)15-s + 1.28e11i·17-s + 2.79e12·19-s + 7.02e12·21-s − 7.47e12i·23-s + (1.78e13 − 6.66e12i)25-s + 1.01e14i·27-s + 1.44e14·29-s − 5.67e13·31-s + ⋯
L(s)  = 1  − 1.84i·3-s + (−0.984 + 0.177i)5-s + 1.04i·7-s − 2.38·9-s − 1.25·11-s − 1.17i·13-s + (0.326 + 1.81i)15-s + 0.263i·17-s + 1.98·19-s + 1.92·21-s − 0.865i·23-s + (0.936 − 0.349i)25-s + 2.55i·27-s + 1.85·29-s − 0.385·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.984 + 0.177i$
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: \((2,\ 80,\ (\ :19/2),\ -0.984 + 0.177i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.120723438\)
\(L(\frac12)\) \(\approx\) \(1.120723438\)
\(L(\frac{21}{2})\) 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 \( 1 + (4.29e6 - 7.75e5i)T \)
good3 \( 1 + 6.27e4iT - 1.16e9T^{2} \)
7 \( 1 - 1.11e8iT - 1.13e16T^{2} \)
11 \( 1 + 9.81e9T + 6.11e19T^{2} \)
13 \( 1 + 4.48e10iT - 1.46e21T^{2} \)
17 \( 1 - 1.28e11iT - 2.39e23T^{2} \)
19 \( 1 - 2.79e12T + 1.97e24T^{2} \)
23 \( 1 + 7.47e12iT - 7.46e25T^{2} \)
29 \( 1 - 1.44e14T + 6.10e27T^{2} \)
31 \( 1 + 5.67e13T + 2.16e28T^{2} \)
37 \( 1 + 2.24e14iT - 6.24e29T^{2} \)
41 \( 1 - 1.34e15T + 4.39e30T^{2} \)
43 \( 1 - 9.67e14iT - 1.08e31T^{2} \)
47 \( 1 + 2.24e15iT - 5.88e31T^{2} \)
53 \( 1 - 3.89e16iT - 5.77e32T^{2} \)
59 \( 1 - 8.51e16T + 4.42e33T^{2} \)
61 \( 1 + 7.94e16T + 8.34e33T^{2} \)
67 \( 1 - 1.91e17iT - 4.95e34T^{2} \)
71 \( 1 + 2.27e17T + 1.49e35T^{2} \)
73 \( 1 + 6.53e17iT - 2.53e35T^{2} \)
79 \( 1 + 2.73e17T + 1.13e36T^{2} \)
83 \( 1 + 9.71e17iT - 2.90e36T^{2} \)
89 \( 1 - 1.09e18T + 1.09e37T^{2} \)
97 \( 1 - 2.10e18iT - 5.60e37T^{2} \)
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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

−10.57514341495159681410298943108, −8.675658630645331621467726168851, −7.908886711813750757705108342493, −7.33575484987297995252577051699, −6.03926514078325587322143698126, −5.16741698614406113309772079322, −2.95278936662023534557266062239, −2.63103756031103990238367451984, −1.07740461491559345153206590464, −0.31597473796962896819139379376, 0.798697764790182835425451665234, 2.94115124925191942677891715405, 3.74012214983972503098207912452, 4.60135854603289304538805014700, 5.30140459460036309650647757011, 7.18206087726345031488640067184, 8.220511232531950637578957691111, 9.419930900431714192302098404674, 10.20809807345704877654139517250, 11.13289657005476078766332123822

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