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Properties

Label 2-1020-1020.779-c0-0-3
Degree $2$
Conductor $1020$
Sign $0.329 - 0.944i$
Analytic cond. $0.509046$
Root an. cond. $0.713474$
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  + (0.195 + 0.980i)2-s + (0.980 + 0.195i)3-s + (−0.923 + 0.382i)4-s + (0.831 − 0.555i)5-s + i·6-s + (−0.555 − 0.831i)8-s + (0.923 + 0.382i)9-s + (0.707 + 0.707i)10-s + (−0.980 + 0.195i)12-s + (0.923 − 0.382i)15-s + (0.707 − 0.707i)16-s + (−0.195 + 0.980i)17-s + (−0.195 + 0.980i)18-s + (−0.707 − 1.70i)19-s + (−0.555 + 0.831i)20-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)2-s + (0.980 + 0.195i)3-s + (−0.923 + 0.382i)4-s + (0.831 − 0.555i)5-s + i·6-s + (−0.555 − 0.831i)8-s + (0.923 + 0.382i)9-s + (0.707 + 0.707i)10-s + (−0.980 + 0.195i)12-s + (0.923 − 0.382i)15-s + (0.707 − 0.707i)16-s + (−0.195 + 0.980i)17-s + (−0.195 + 0.980i)18-s + (−0.707 − 1.70i)19-s + (−0.555 + 0.831i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(0.509046\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :0),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.550703322\)
\(L(\frac12)\) \(\approx\) \(1.550703322\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.195 - 0.980i)T \)
3 \( 1 + (-0.980 - 0.195i)T \)
5 \( 1 + (-0.831 + 0.555i)T \)
17 \( 1 + (0.195 - 0.980i)T \)
good7 \( 1 + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (1.38 - 0.275i)T + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.324 - 1.63i)T + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.382 - 0.923i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.785 - 0.785i)T - iT^{2} \)
53 \( 1 + (1.81 - 0.750i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.360 + 0.149i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.382 - 0.923i)T^{2} \)
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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

−9.914437345023217708602727437320, −9.253226783918015551080970532415, −8.581644244512908938810552429866, −8.022840247702487386682052027749, −6.88395186059148679973911048129, −6.19232696400999646484341058219, −5.02506316804037485091542912762, −4.38067785254603877833570760288, −3.22531034581176292153645940059, −1.85370632273633537598927648189, 1.74720524987380151883600201197, 2.44062101059661745477876156856, 3.49006000784557768419573654670, 4.34787266416952072549575463791, 5.67239170408535578623451804683, 6.49129228509799258112241420954, 7.75589832340586626373278949816, 8.458560745678633358491248936088, 9.581528064753861057980760635193, 9.806969192491339103771231120150

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