Properties

Label 2-80-80.29-c1-0-5
Degree $2$
Conductor $80$
Sign $0.418 + 0.908i$
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 + (1.90 + 1.17i)5-s + (−3.68 + 0.547i)6-s − 3.61·7-s + (0.949 − 2.66i)8-s − 3.92i·9-s + (−1.17 − 2.93i)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 + (0.851 + 0.524i)5-s + (−1.50 + 0.223i)6-s − 1.36·7-s + (0.335 − 0.941i)8-s − 1.30i·9-s + (−0.372 − 0.928i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.418 + 0.908i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.418 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753169 - 0.482356i\)
\(L(\frac12)\) \(\approx\) \(0.753169 - 0.482356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.90 - 1.17i)T \)
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

−13.82847576656962913252801346890, −13.05694459629174505969190781470, −12.33986047545825210035673203978, −10.64967745930500888170251622803, −9.502962181567309642591554514243, −8.822451384705498606336402696872, −7.22366820475527320783119336313, −6.61696788193326706845104818642, −3.21651712146688656948704997612, −2.13047183919042619871583100321, 2.85960730156526136200675067883, 4.97694382234088233112454000718, 6.44448964347847025596203809124, 8.105340432036831543114380793582, 9.302086819566607319478063112871, 9.670302013733668688465038738594, 10.54508363402514642825563356567, 12.74821346119324110168873958806, 13.85317334627600983407283452254, 14.84730664373456822450275023387

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