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Properties

Label 2-80-80.27-c1-0-7
Degree $2$
Conductor $80$
Sign $-0.811 + 0.584i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 2·3-s − 2i·4-s + (−2 + i)5-s + (2 − 2i)6-s + (−3 − 3i)7-s + (2 + 2i)8-s + 9-s + (1 − 3i)10-s + (−1 + i)11-s + 4i·12-s + 2i·13-s + 6·14-s + (4 − 2i)15-s − 4·16-s + (1 + i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.15·3-s − i·4-s + (−0.894 + 0.447i)5-s + (0.816 − 0.816i)6-s + (−1.13 − 1.13i)7-s + (0.707 + 0.707i)8-s + 0.333·9-s + (0.316 − 0.948i)10-s + (−0.301 + 0.301i)11-s + 1.15i·12-s + 0.554i·13-s + 1.60·14-s + (1.03 − 0.516i)15-s − 16-s + (0.242 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.811 + 0.584i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ -0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 + (2 - i)T \)
good3 \( 1 + 2T + 3T^{2} \)
7 \( 1 + (3 + 3i)T + 7iT^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (7 + 7i)T + 29iT^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (7 - 7i)T - 47iT^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + (-3 - 3i)T + 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (11 + 11i)T + 97iT^{2} \)
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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

−14.21442938078161757019809759424, −12.80830581494530963976800378598, −11.43204301764673511085392549231, −10.61649948138300935650475551650, −9.699859526262278865980790027689, −7.88638268394394799902523829232, −6.88805760583972888980449229505, −5.99255617960231670738492661515, −4.16929381792922326649904005931, 0, 3.19744395278082895586299779894, 5.18013618333723223342449259876, 6.69179242865704678912310909522, 8.294765114665058736602328411165, 9.307682822426627168383887243707, 10.68161814271764683129864823345, 11.56026651288003495969755770482, 12.43225073815099575731022978335, 12.96317739991569446309415491808

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