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Properties

Label 2-1020-1020.299-c0-0-3
Degree $2$
Conductor $1020$
Sign $-0.423 + 0.906i$
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  + (0.831 − 0.555i)2-s + (0.555 − 0.831i)3-s + (0.382 − 0.923i)4-s + (−0.980 + 0.195i)5-s − i·6-s + (−0.195 − 0.980i)8-s + (−0.382 − 0.923i)9-s + (−0.707 + 0.707i)10-s + (−0.555 − 0.831i)12-s + (−0.382 + 0.923i)15-s + (−0.707 − 0.707i)16-s + (−0.831 − 0.555i)17-s + (−0.831 − 0.555i)18-s + (0.707 + 0.292i)19-s + (−0.195 + 0.980i)20-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)2-s + (0.555 − 0.831i)3-s + (0.382 − 0.923i)4-s + (−0.980 + 0.195i)5-s − i·6-s + (−0.195 − 0.980i)8-s + (−0.382 − 0.923i)9-s + (−0.707 + 0.707i)10-s + (−0.555 − 0.831i)12-s + (−0.382 + 0.923i)15-s + (−0.707 − 0.707i)16-s + (−0.831 − 0.555i)17-s + (−0.831 − 0.555i)18-s + (0.707 + 0.292i)19-s + (−0.195 + 0.980i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.423 + 0.906i$
Analytic conductor: \(0.509046\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :0),\ -0.423 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.598990445\)
\(L(\frac12)\) \(\approx\) \(1.598990445\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 0.555i)T \)
3 \( 1 + (-0.555 + 0.831i)T \)
5 \( 1 + (0.980 - 0.195i)T \)
17 \( 1 + (0.831 + 0.555i)T \)
good7 \( 1 + (0.923 + 0.382i)T^{2} \)
11 \( 1 + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.785 - 1.17i)T + (-0.382 + 0.923i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (-0.923 - 0.382i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.275 - 0.275i)T + iT^{2} \)
53 \( 1 + (-0.425 + 1.02i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.923 + 0.382i)T^{2} \)
79 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.923 - 0.382i)T^{2} \)
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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

−9.962151020062601418927534252496, −9.053544395045903542663182384059, −8.137173748984588445107503826433, −7.14219218697451247827905996449, −6.70621551764730698058130446459, −5.46973551704015787589079570648, −4.40898767480524850426026504466, −3.37146976327067581398487489892, −2.70410768000419378180703777729, −1.22664354114791953041737218556, 2.56471305480360113165921321398, 3.44920010401175036984656407169, 4.44724761054085639217957645820, 4.80208579409195928110052882611, 6.09317279627518405377900994854, 7.10037224352414429964382687665, 8.005343783562708440794315858800, 8.526708574412993516098818965542, 9.358119416419886191232412797054, 10.60689526337316620396585205166

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