Properties

Label 2-80-80.53-c10-0-93
Degree $2$
Conductor $80$
Sign $-0.805 - 0.592i$
Analytic cond. $50.8285$
Root an. cond. $7.12941$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−29.2 − 12.9i)2-s − 227.·3-s + (690. + 756. i)4-s + (2.87e3 − 1.22e3i)5-s + (6.64e3 + 2.93e3i)6-s + (−1.16e4 − 1.16e4i)7-s + (−1.04e4 − 3.10e4i)8-s − 7.51e3·9-s + (−9.99e4 − 1.30e3i)10-s + (6.74e4 + 6.74e4i)11-s + (−1.56e5 − 1.71e5i)12-s + 1.36e5·13-s + (1.90e5 + 4.91e5i)14-s + (−6.52e5 + 2.77e5i)15-s + (−9.48e4 + 1.04e6i)16-s + (−7.16e5 + 7.16e5i)17-s + ⋯
L(s)  = 1  + (−0.914 − 0.403i)2-s − 0.934·3-s + (0.674 + 0.738i)4-s + (0.920 − 0.391i)5-s + (0.854 + 0.376i)6-s + (−0.692 − 0.692i)7-s + (−0.319 − 0.947i)8-s − 0.127·9-s + (−0.999 − 0.0130i)10-s + (0.418 + 0.418i)11-s + (−0.629 − 0.689i)12-s + 0.366·13-s + (0.354 + 0.912i)14-s + (−0.859 + 0.365i)15-s + (−0.0904 + 0.995i)16-s + (−0.504 + 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ -0.805 - 0.592i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0664724 + 0.202437i\)
\(L(\frac12)\) \(\approx\) \(0.0664724 + 0.202437i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (29.2 + 12.9i)T \)
5 \( 1 + (-2.87e3 + 1.22e3i)T \)
good3 \( 1 + 227.T + 5.90e4T^{2} \)
7 \( 1 + (1.16e4 + 1.16e4i)T + 2.82e8iT^{2} \)
11 \( 1 + (-6.74e4 - 6.74e4i)T + 2.59e10iT^{2} \)
13 \( 1 - 1.36e5T + 1.37e11T^{2} \)
17 \( 1 + (7.16e5 - 7.16e5i)T - 2.01e12iT^{2} \)
19 \( 1 + (1.93e6 + 1.93e6i)T + 6.13e12iT^{2} \)
23 \( 1 + (-7.82e6 + 7.82e6i)T - 4.14e13iT^{2} \)
29 \( 1 + (2.24e7 + 2.24e7i)T + 4.20e14iT^{2} \)
31 \( 1 - 6.83e6T + 8.19e14T^{2} \)
37 \( 1 - 8.63e7T + 4.80e15T^{2} \)
41 \( 1 - 2.13e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.88e8iT - 2.16e16T^{2} \)
47 \( 1 + (4.06e7 - 4.06e7i)T - 5.25e16iT^{2} \)
53 \( 1 - 9.52e7iT - 1.74e17T^{2} \)
59 \( 1 + (-3.32e8 + 3.32e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (7.28e8 - 7.28e8i)T - 7.13e17iT^{2} \)
67 \( 1 + 1.33e9iT - 1.82e18T^{2} \)
71 \( 1 - 5.02e8iT - 3.25e18T^{2} \)
73 \( 1 + (9.57e8 - 9.57e8i)T - 4.29e18iT^{2} \)
79 \( 1 + 8.06e8iT - 9.46e18T^{2} \)
83 \( 1 + 6.43e9T + 1.55e19T^{2} \)
89 \( 1 + 5.67e9T + 3.11e19T^{2} \)
97 \( 1 + (9.72e9 - 9.72e9i)T - 7.37e19iT^{2} \)
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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

−11.34258130125686643917106493633, −10.56895945757195166829500128041, −9.577493937337826101384617056697, −8.580555041005701967025162142887, −6.80959842854072923124851408886, −6.15230739107871678053496705001, −4.38299950622832710640117539046, −2.56201620330658139919169191461, −1.07758810264733490884804187097, −0.10102931512547005100709937586, 1.39411494940001983009807714276, 2.86549997131643934000436545120, 5.47356806291228384029827503823, 6.05979573788137434382640744959, 6.98201957110243312289242993263, 8.794573586393033518547542586441, 9.539267918606407111760651835037, 10.79018391950937029195129448310, 11.44563181249888461393516116983, 12.84604327911470063187963053056

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