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Properties

Label 2-80-80.69-c1-0-3
Degree $2$
Conductor $80$
Sign $0.686 + 0.727i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 − 0.238i)2-s + (−0.183 − 0.183i)3-s + (1.88 + 0.666i)4-s + (−0.569 − 2.16i)5-s + (0.212 + 0.300i)6-s + 3.84·7-s + (−2.46 − 1.37i)8-s − 2.93i·9-s + (0.277 + 3.15i)10-s + (1.60 + 1.60i)11-s + (−0.224 − 0.469i)12-s + (−1.80 − 1.80i)13-s + (−5.36 − 0.919i)14-s + (−0.292 + 0.502i)15-s + (3.11 + 2.51i)16-s + 4.93i·17-s + ⋯
L(s)  = 1  + (−0.985 − 0.168i)2-s + (−0.106 − 0.106i)3-s + (0.942 + 0.333i)4-s + (−0.254 − 0.966i)5-s + (0.0866 + 0.122i)6-s + 1.45·7-s + (−0.873 − 0.487i)8-s − 0.977i·9-s + (0.0877 + 0.996i)10-s + (0.482 + 0.482i)11-s + (−0.0647 − 0.135i)12-s + (−0.501 − 0.501i)13-s + (−1.43 − 0.245i)14-s + (−0.0755 + 0.129i)15-s + (0.778 + 0.628i)16-s + 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611376 - 0.263593i\)
\(L(\frac12)\) \(\approx\) \(0.611376 - 0.263593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.238i)T \)
5 \( 1 + (0.569 + 2.16i)T \)
good3 \( 1 + (0.183 + 0.183i)T + 3iT^{2} \)
7 \( 1 - 3.84T + 7T^{2} \)
11 \( 1 + (-1.60 - 1.60i)T + 11iT^{2} \)
13 \( 1 + (1.80 + 1.80i)T + 13iT^{2} \)
17 \( 1 - 4.93iT - 17T^{2} \)
19 \( 1 + (4.77 - 4.77i)T - 19iT^{2} \)
23 \( 1 - 0.134T + 23T^{2} \)
29 \( 1 + (-2.17 + 2.17i)T - 29iT^{2} \)
31 \( 1 - 2.26T + 31T^{2} \)
37 \( 1 + (-4.35 + 4.35i)T - 37iT^{2} \)
41 \( 1 - 3.34iT - 41T^{2} \)
43 \( 1 + (2.70 - 2.70i)T - 43iT^{2} \)
47 \( 1 - 7.03iT - 47T^{2} \)
53 \( 1 + (3.40 - 3.40i)T - 53iT^{2} \)
59 \( 1 + (0.107 + 0.107i)T + 59iT^{2} \)
61 \( 1 + (3.46 - 3.46i)T - 61iT^{2} \)
67 \( 1 + (1.91 + 1.91i)T + 67iT^{2} \)
71 \( 1 + 9.32iT - 71T^{2} \)
73 \( 1 - 9.82T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (-8.80 - 8.80i)T + 83iT^{2} \)
89 \( 1 - 1.12iT - 89T^{2} \)
97 \( 1 + 6.10iT - 97T^{2} \)
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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

−14.69382155198717277260617986301, −12.60178167392301467873025832574, −12.09586734539249555343720353985, −10.96275283963623095521318148068, −9.675256026200400757376880337179, −8.484601455951471830256080357010, −7.80916025777719112160372191063, −6.10215074392814496239830110346, −4.24506280589737075028477561108, −1.52746321359538754853897119772, 2.34958294078332596318156431949, 4.90935424394407053030088576251, 6.71681848073864374418711334772, 7.71634317272849894907011604440, 8.758610135004346994881694249837, 10.26325593145792112656535739209, 11.23999471266225716394877197490, 11.62936383850112906305409627999, 13.89064567601405946454404916991, 14.66130554050946555364748934375

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