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Properties

Label 2-1020-1020.419-c0-0-3
Degree $2$
Conductor $1020$
Sign $-0.434 + 0.900i$
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  + (0.555 − 0.831i)2-s + (0.980 − 0.195i)3-s + (−0.382 − 0.923i)4-s + (−0.831 − 0.555i)5-s + (0.382 − 0.923i)6-s + (−0.980 − 0.195i)8-s + (0.923 − 0.382i)9-s + (−0.923 + 0.382i)10-s + (−0.555 − 0.831i)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 + (−0.195 + 0.980i)20-s + ⋯
L(s)  = 1  + (0.555 − 0.831i)2-s + (0.980 − 0.195i)3-s + (−0.382 − 0.923i)4-s + (−0.831 − 0.555i)5-s + (0.382 − 0.923i)6-s + (−0.980 − 0.195i)8-s + (0.923 − 0.382i)9-s + (−0.923 + 0.382i)10-s + (−0.555 − 0.831i)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 + (−0.195 + 0.980i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(0.509046\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :0),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.530317859\)
\(L(\frac12)\) \(\approx\) \(1.530317859\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.555 + 0.831i)T \)
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

−9.947093879928456962834608338016, −8.951802858265369292317195692398, −8.524293903291721334168234431187, −7.49255069390364569606032116998, −6.52371274705024350939751310666, −5.19775925720123173551416883133, −4.30766274787035329563234305992, −3.56531610784631919943892544036, −2.58260026392467077265575628889, −1.25255579459130105101455211483, 2.43412095366924676402637839236, 3.57688861118725551771499200141, 3.99231477886807120984625546153, 5.17559139379194055681461016526, 6.28965180539425115974385887155, 7.27876318363164597401221764791, 7.906748580642726339731207481936, 8.304551686261845238495847621379, 9.559558056126246165551194723746, 10.08010898260901117519376903051

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