Properties

Label 2-1020-1020.719-c0-0-2
Degree $2$
Conductor $1020$
Sign $0.0318 + 0.999i$
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  + (−0.980 − 0.195i)2-s + (0.831 − 0.555i)3-s + (0.923 + 0.382i)4-s + (0.195 − 0.980i)5-s + (−0.923 + 0.382i)6-s + (−0.831 − 0.555i)8-s + (0.382 − 0.923i)9-s + (−0.382 + 0.923i)10-s + (0.980 − 0.195i)12-s + (−0.382 − 0.923i)15-s + (0.707 + 0.707i)16-s + (−0.555 − 0.831i)17-s + (−0.555 + 0.831i)18-s + (−0.707 + 0.292i)19-s + (0.555 − 0.831i)20-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (0.831 − 0.555i)3-s + (0.923 + 0.382i)4-s + (0.195 − 0.980i)5-s + (−0.923 + 0.382i)6-s + (−0.831 − 0.555i)8-s + (0.382 − 0.923i)9-s + (−0.382 + 0.923i)10-s + (0.980 − 0.195i)12-s + (−0.382 − 0.923i)15-s + (0.707 + 0.707i)16-s + (−0.555 − 0.831i)17-s + (−0.555 + 0.831i)18-s + (−0.707 + 0.292i)19-s + (0.555 − 0.831i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0318 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0318 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.0318 + 0.999i$
Analytic conductor: \(0.509046\)
Root analytic conductor: \(0.713474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1020,\ (\ :0),\ 0.0318 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8947267501\)
\(L(\frac12)\) \(\approx\) \(0.8947267501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 + 0.195i)T \)
3 \( 1 + (-0.831 + 0.555i)T \)
5 \( 1 + (-0.195 + 0.980i)T \)
17 \( 1 + (0.555 + 0.831i)T \)
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 \( 1 + (-1.17 - 0.785i)T + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
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 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
53 \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.382 + 0.0761i)T + (0.923 - 0.382i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.425 + 1.02i)T + (-0.707 + 0.707i)T^{2} \)
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

−9.545856403433069239910014463880, −9.100947903926501685788875206398, −8.461629542066695047661490603108, −7.66372983802889621545610157792, −6.90929403191195625085200114808, −5.93541174292366058735852732451, −4.56738317511827721671786177130, −3.28557591359139390729466107933, −2.20292262920009732387955782383, −1.12030872322884848244546111926, 2.01165920898450086482298397064, 2.80570627200465257618672620433, 3.92177350249309932517691516364, 5.28519402318249599625034599298, 6.53588015991020601016900882326, 7.02894355597651003763381521135, 8.138238707285618857051042547790, 8.652571180034436484062322802813, 9.540938961510794858425689600251, 10.25417710104045950447333776809

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