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Properties

Label 2-80-16.13-c1-0-2
Degree $2$
Conductor $80$
Sign $0.321 - 0.946i$
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  + (0.562 + 1.29i)2-s + (0.209 + 0.209i)3-s + (−1.36 + 1.45i)4-s + (0.707 − 0.707i)5-s + (−0.154 + 0.389i)6-s + 1.73i·7-s + (−2.66 − 0.952i)8-s − 2.91i·9-s + (1.31 + 0.519i)10-s + (0.505 − 0.505i)11-s + (−0.592 + 0.0194i)12-s + (−1.88 − 1.88i)13-s + (−2.25 + 0.977i)14-s + 0.296·15-s + (−0.262 − 3.99i)16-s + 4.53·17-s + ⋯
L(s)  = 1  + (0.397 + 0.917i)2-s + (0.120 + 0.120i)3-s + (−0.683 + 0.729i)4-s + (0.316 − 0.316i)5-s + (−0.0628 + 0.159i)6-s + 0.656i·7-s + (−0.941 − 0.336i)8-s − 0.970i·9-s + (0.415 + 0.164i)10-s + (0.152 − 0.152i)11-s + (−0.171 + 0.00561i)12-s + (−0.523 − 0.523i)13-s + (−0.602 + 0.261i)14-s + 0.0765·15-s + (−0.0655 − 0.997i)16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910638 + 0.652648i\)
\(L(\frac12)\) \(\approx\) \(0.910638 + 0.652648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.562 - 1.29i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.209 - 0.209i)T + 3iT^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + (-0.505 + 0.505i)T - 11iT^{2} \)
13 \( 1 + (1.88 + 1.88i)T + 13iT^{2} \)
17 \( 1 - 4.53T + 17T^{2} \)
19 \( 1 + (3.22 + 3.22i)T + 19iT^{2} \)
23 \( 1 - 8.85iT - 23T^{2} \)
29 \( 1 + (2.44 + 2.44i)T + 29iT^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 + (5.35 - 5.35i)T - 37iT^{2} \)
41 \( 1 - 10.0iT - 41T^{2} \)
43 \( 1 + (2.10 - 2.10i)T - 43iT^{2} \)
47 \( 1 - 4.32T + 47T^{2} \)
53 \( 1 + (1.37 - 1.37i)T - 53iT^{2} \)
59 \( 1 + (-6.64 + 6.64i)T - 59iT^{2} \)
61 \( 1 + (-5.26 - 5.26i)T + 61iT^{2} \)
67 \( 1 + (10.5 + 10.5i)T + 67iT^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + 6.63iT - 73T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + (-9.15 - 9.15i)T + 83iT^{2} \)
89 \( 1 - 3.23iT - 89T^{2} \)
97 \( 1 - 1.94T + 97T^{2} \)
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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

−14.92799911556472090816870719680, −13.64836172894177716997041997252, −12.63523425561765253982061124271, −11.76233078115189870350164928693, −9.719099882024863277772482871803, −8.935514093002883866758704993845, −7.64420686629694987094620622626, −6.19630405849260286173486799408, −5.16220912860181285299764065021, −3.40262913038877110843840701119, 2.15694204486794891944584670387, 4.00474697638882823292828938126, 5.46835453361246655882758634900, 7.15866525257726208591423397397, 8.743489391040442548666316258199, 10.23417910637396873061323568057, 10.68555411513024738462666858584, 12.14337761863172029024201584505, 13.03219913014797193643474418612, 14.27371398753111586618978286863

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