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Properties

Label 2-1020-1020.719-c0-0-3
Degree $2$
Conductor $1020$
Sign $-0.684 + 0.729i$
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 + (−0.831 + 0.555i)3-s + (−0.382 − 0.923i)4-s + (0.195 − 0.980i)5-s + i·6-s + (−0.980 − 0.195i)8-s + (0.382 − 0.923i)9-s + (−0.707 − 0.707i)10-s + (0.831 + 0.555i)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 + (−0.980 + 0.195i)20-s + ⋯
L(s)  = 1  + (0.555 − 0.831i)2-s + (−0.831 + 0.555i)3-s + (−0.382 − 0.923i)4-s + (0.195 − 0.980i)5-s + i·6-s + (−0.980 − 0.195i)8-s + (0.382 − 0.923i)9-s + (−0.707 − 0.707i)10-s + (0.831 + 0.555i)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 + (−0.980 + 0.195i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-0.684 + 0.729i$
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: \((2,\ 1020,\ (\ :0),\ -0.684 + 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8982162879\)
\(L(\frac12)\) \(\approx\) \(0.8982162879\)
\(L(1)\) 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 \( 1 + (0.831 - 0.555i)T \)
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

−9.983164332456676525852075270309, −9.300492602438264112090953302730, −8.611310880318213267238071991150, −7.09171665240944436966682821607, −5.95930860679563855021117740352, −5.35623568017140653697581189777, −4.53240570346326620991517287198, −3.85409077319336920690730657984, −2.34288528506834513253007846185, −0.791382404187776367628833591577, 2.09147345166363936232908207961, 3.45352036533856374001222405229, 4.49271507591971116875925348337, 5.79247789499220214145037907414, 6.00091385465487178881580206397, 7.08289968580993220265961982380, 7.52315846654376566764529958145, 8.472376954145562505984774106932, 9.706084758567560227486080527293, 10.56588776492994103782758321581

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