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Properties

Label 2-80-80.69-c1-0-9
Degree $2$
Conductor $80$
Sign $0.0356 + 0.999i$
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  + (1.13 − 0.841i)2-s + (−1.86 − 1.86i)3-s + (0.582 − 1.91i)4-s + (−1.17 + 1.90i)5-s + (−3.68 − 0.547i)6-s + 3.61·7-s + (−0.949 − 2.66i)8-s + 3.92i·9-s + (0.271 + 3.15i)10-s + (−0.0947 − 0.0947i)11-s + (−4.64 + 2.47i)12-s + (2.59 + 2.59i)13-s + (4.10 − 3.04i)14-s + (5.72 − 1.36i)15-s + (−3.32 − 2.22i)16-s + 1.89i·17-s + ⋯
L(s)  = 1  + (0.803 − 0.595i)2-s + (−1.07 − 1.07i)3-s + (0.291 − 0.956i)4-s + (−0.524 + 0.851i)5-s + (−1.50 − 0.223i)6-s + 1.36·7-s + (−0.335 − 0.941i)8-s + 1.30i·9-s + (0.0858 + 0.996i)10-s + (−0.0285 − 0.0285i)11-s + (−1.34 + 0.714i)12-s + (0.719 + 0.719i)13-s + (1.09 − 0.813i)14-s + (1.47 − 0.351i)15-s + (−0.830 − 0.556i)16-s + 0.460i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0356 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0356 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.0356 + 0.999i$
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: \((2,\ 80,\ (\ :1/2),\ 0.0356 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772669 - 0.745583i\)
\(L(\frac12)\) \(\approx\) \(0.772669 - 0.745583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 0.841i)T \)
5 \( 1 + (1.17 - 1.90i)T \)
good3 \( 1 + (1.86 + 1.86i)T + 3iT^{2} \)
7 \( 1 - 3.61T + 7T^{2} \)
11 \( 1 + (0.0947 + 0.0947i)T + 11iT^{2} \)
13 \( 1 + (-2.59 - 2.59i)T + 13iT^{2} \)
17 \( 1 - 1.89iT - 17T^{2} \)
19 \( 1 + (2.16 - 2.16i)T - 19iT^{2} \)
23 \( 1 + 5.08T + 23T^{2} \)
29 \( 1 + (-1.25 + 1.25i)T - 29iT^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \)
41 \( 1 - 8.52iT - 41T^{2} \)
43 \( 1 + (-1.61 + 1.61i)T - 43iT^{2} \)
47 \( 1 + 2.53iT - 47T^{2} \)
53 \( 1 + (-5.67 + 5.67i)T - 53iT^{2} \)
59 \( 1 + (7.81 + 7.81i)T + 59iT^{2} \)
61 \( 1 + (-3.46 + 3.46i)T - 61iT^{2} \)
67 \( 1 + (-6.29 - 6.29i)T + 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 - 1.13T + 79T^{2} \)
83 \( 1 + (3.75 + 3.75i)T + 83iT^{2} \)
89 \( 1 + 3.98iT - 89T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−14.05794134165821605915364406931, −12.85581539311948702786602787864, −11.67506511313990209151599122266, −11.41864009601521384996493047447, −10.43270184290162603071521968431, −8.053540303558593094235578730035, −6.75419671620032411042743437130, −5.79531599990083030304929734926, −4.20043566813844357610118406853, −1.80668676801464873908425654601, 4.10604551183961593520974466299, 4.91890419466377989052123064061, 5.82108652583974171396891983396, 7.73951864899361091134915922247, 8.815593561223988286420934456126, 10.70570499765061518694295568104, 11.51668240227791420789817771715, 12.31609816488414275459646529925, 13.68470753231209521201716079442, 15.01126885590924162123683135856

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