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Properties

Label 2-80-16.5-c1-0-0
Degree $2$
Conductor $80$
Sign $-0.943 - 0.332i$
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  + (−0.257 + 1.39i)2-s + (−1.66 + 1.66i)3-s + (−1.86 − 0.715i)4-s + (−0.707 − 0.707i)5-s + (−1.88 − 2.74i)6-s + 2.89i·7-s + (1.47 − 2.41i)8-s − 2.53i·9-s + (1.16 − 0.801i)10-s + (1.84 + 1.84i)11-s + (4.29 − 1.91i)12-s + (−3.08 + 3.08i)13-s + (−4.02 − 0.744i)14-s + 2.35·15-s + (2.97 + 2.67i)16-s + 7.29·17-s + ⋯
L(s)  = 1  + (−0.181 + 0.983i)2-s + (−0.960 + 0.960i)3-s + (−0.933 − 0.357i)4-s + (−0.316 − 0.316i)5-s + (−0.769 − 1.11i)6-s + 1.09i·7-s + (0.521 − 0.853i)8-s − 0.845i·9-s + (0.368 − 0.253i)10-s + (0.556 + 0.556i)11-s + (1.24 − 0.553i)12-s + (−0.854 + 0.854i)13-s + (−1.07 − 0.198i)14-s + 0.607·15-s + (0.744 + 0.667i)16-s + 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.943 - 0.332i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ -0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0928056 + 0.542761i\)
\(L(\frac12)\) \(\approx\) \(0.0928056 + 0.542761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.257 - 1.39i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.66 - 1.66i)T - 3iT^{2} \)
7 \( 1 - 2.89iT - 7T^{2} \)
11 \( 1 + (-1.84 - 1.84i)T + 11iT^{2} \)
13 \( 1 + (3.08 - 3.08i)T - 13iT^{2} \)
17 \( 1 - 7.29T + 17T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 + 4.60iT - 23T^{2} \)
29 \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \)
31 \( 1 - 2.06T + 31T^{2} \)
37 \( 1 + (1.17 + 1.17i)T + 37iT^{2} \)
41 \( 1 - 4.61iT - 41T^{2} \)
43 \( 1 + (-3.03 - 3.03i)T + 43iT^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + (-2.73 - 2.73i)T + 53iT^{2} \)
59 \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \)
61 \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \)
67 \( 1 + (-8.24 + 8.24i)T - 67iT^{2} \)
71 \( 1 + 3.25iT - 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 + 0.113T + 79T^{2} \)
83 \( 1 + (-9.76 + 9.76i)T - 83iT^{2} \)
89 \( 1 + 3.74iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−15.04916599349838158583265460202, −14.37562126627049743189675909840, −12.45321322321811966929187599860, −11.77723480016359803312400669645, −10.10380359667736535198896798707, −9.403617648452073117310403902371, −8.069692462943350046187945433367, −6.43794282742513278656978756327, −5.29175108229792716368871042827, −4.35940089717309867763327357861, 0.931318203892303528770484283224, 3.47612250454271593007340283161, 5.34867073685228489067700649992, 7.04414505515548922092027906268, 8.009191967631877203615173692207, 9.902504592126109091657337519928, 10.82141117770155749271078176380, 11.81293069923267595563058990473, 12.50592131344513304143180107662, 13.57834604514510988001453272074

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