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Properties

Label 2-820-820.503-c0-0-0
Degree $2$
Conductor $820$
Sign $-0.621 + 0.783i$
Analytic cond. $0.409233$
Root an. cond. $0.639713$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.156 − 0.987i)5-s + (−0.951 − 0.309i)8-s + (−0.707 + 0.707i)9-s + (−0.891 − 0.453i)10-s + (0.465 − 1.93i)13-s + (−0.809 + 0.587i)16-s + (0.355 + 0.303i)17-s + (0.156 + 0.987i)18-s + (−0.891 + 0.453i)20-s + (−0.951 + 0.309i)25-s + (−1.29 − 1.51i)26-s + (0.763 + 0.0600i)29-s + i·32-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.156 − 0.987i)5-s + (−0.951 − 0.309i)8-s + (−0.707 + 0.707i)9-s + (−0.891 − 0.453i)10-s + (0.465 − 1.93i)13-s + (−0.809 + 0.587i)16-s + (0.355 + 0.303i)17-s + (0.156 + 0.987i)18-s + (−0.891 + 0.453i)20-s + (−0.951 + 0.309i)25-s + (−1.29 − 1.51i)26-s + (0.763 + 0.0600i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.621 + 0.783i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ -0.621 + 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.139280493\)
\(L(\frac12)\) \(\approx\) \(1.139280493\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (0.156 + 0.987i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.453 + 0.891i)T^{2} \)
11 \( 1 + (-0.156 - 0.987i)T^{2} \)
13 \( 1 + (-0.465 + 1.93i)T + (-0.891 - 0.453i)T^{2} \)
17 \( 1 + (-0.355 - 0.303i)T + (0.156 + 0.987i)T^{2} \)
19 \( 1 + (-0.453 - 0.891i)T^{2} \)
23 \( 1 + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.763 - 0.0600i)T + (0.987 + 0.156i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-1.69 - 0.863i)T + (0.587 + 0.809i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.453 + 0.891i)T^{2} \)
53 \( 1 + (-0.678 - 0.794i)T + (-0.156 + 0.987i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
67 \( 1 + (-0.987 - 0.156i)T^{2} \)
71 \( 1 + (0.156 + 0.987i)T^{2} \)
73 \( 1 - 1.78T + T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.431 - 1.79i)T + (-0.891 + 0.453i)T^{2} \)
97 \( 1 + (-0.119 + 1.51i)T + (-0.987 - 0.156i)T^{2} \)
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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

−10.30339829054496788343828145725, −9.495002328410961635080694395479, −8.322208271839732593499339538144, −8.025874242488265208672623127488, −6.23165750596006759386394771900, −5.41922758073780701262168987643, −4.83422050677532442691145861019, −3.59911204531378900137908253202, −2.61356029375699513854250328646, −1.04689021898343795788012576841, 2.50244897645930169507999669769, 3.58690649192093435321330140335, 4.35985999708406966406369834058, 5.71661730960459787965256244903, 6.49270040652828985876529266441, 7.00420311743385209293469518731, 8.029119924224715510971607201618, 8.966358386943615885105269511186, 9.639929586648027619105987888880, 11.01672450457358013592820589574

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