Properties

Degree $2$
Conductor $80$
Sign $-0.804 + 0.593i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.8i·3-s + (−45 + 33.1i)5-s + 59.6i·7-s − 153·9-s − 252·11-s − 119. i·13-s + (−660 − 895. i)15-s − 689. i·17-s + 220·19-s − 1.18e3·21-s − 2.43e3i·23-s + (924. − 2.98e3i)25-s + 1.79e3i·27-s − 6.93e3·29-s − 6.75e3·31-s + ⋯
L(s)  = 1  + 1.27i·3-s + (−0.804 + 0.593i)5-s + 0.460i·7-s − 0.629·9-s − 0.627·11-s − 0.195i·13-s + (−0.757 − 1.02i)15-s − 0.578i·17-s + 0.139·19-s − 0.587·21-s − 0.959i·23-s + (0.295 − 0.955i)25-s + 0.472i·27-s − 1.53·29-s − 1.26·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.804 + 0.593i$
Motivic weight: \(5\)
Character: $\chi_{80} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.804 + 0.593i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.183205 - 0.557366i\)
\(L(\frac12)\) \(\approx\) \(0.183205 - 0.557366i\)
\(L(\frac{7}{2})\) 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 \( 1 + (45 - 33.1i)T \)
good3 \( 1 - 19.8iT - 243T^{2} \)
7 \( 1 - 59.6iT - 1.68e4T^{2} \)
11 \( 1 + 252T + 1.61e5T^{2} \)
13 \( 1 + 119. iT - 3.71e5T^{2} \)
17 \( 1 + 689. iT - 1.41e6T^{2} \)
19 \( 1 - 220T + 2.47e6T^{2} \)
23 \( 1 + 2.43e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.93e3T + 2.05e7T^{2} \)
31 \( 1 + 6.75e3T + 2.86e7T^{2} \)
37 \( 1 - 1.39e4iT - 6.93e7T^{2} \)
41 \( 1 + 198T + 1.15e8T^{2} \)
43 \( 1 - 417. iT - 1.47e8T^{2} \)
47 \( 1 - 1.05e4iT - 2.29e8T^{2} \)
53 \( 1 + 5.82e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.46e4T + 7.14e8T^{2} \)
61 \( 1 + 5.69e3T + 8.44e8T^{2} \)
67 \( 1 - 4.36e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.33e4T + 1.80e9T^{2} \)
73 \( 1 - 7.09e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.19e4T + 3.07e9T^{2} \)
83 \( 1 - 6.18e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.99e3T + 5.58e9T^{2} \)
97 \( 1 + 1.01e5iT - 8.58e9T^{2} \)
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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

−14.49752594411039763082254060543, −12.90983674890392720471875134896, −11.58005729088650790866170375174, −10.73355844752777835638069157217, −9.770690845478940023450445049360, −8.540671969408024928511524549821, −7.20400959014858528732836540970, −5.43403235251834930132038685870, −4.16887342677639859435482830729, −2.90215463686548024180531290082, 0.24545829951886585968100040249, 1.72972365910079842681397333888, 3.83913336687543553481393853115, 5.58315848127763899319032472826, 7.25338803591288837699092310252, 7.75938724834058794437686308689, 9.095376398425732577586925992651, 10.81171708991485132610084415289, 11.90777150247344623859242650377, 12.85937452426659599286633648871

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