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Properties

Label 2-1020-1020.719-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(1.393006357\)
\(L(\frac12)\) \(\approx\) \(1.393006357\)
\(L(1)\) 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

−10.42838721591049964677041528727, −10.11138911803217518531250207935, −8.536454757439731719043665970438, −7.58075056128046678575069543776, −6.58803051791276888333527960255, −6.13956736436684183346932252686, −5.23346318845590949606347507491, −4.08918972293917786946354491458, −3.57164697908620452858244888482, −2.18181223939938870383164631483, 1.16327607194070429750335500061, 2.41897542541334512804867164128, 3.99150348531234860620758476215, 4.75844902709527971820648069570, 5.60358033563871663258481663675, 6.20736447058357620356954856787, 7.37100574493524212979384405984, 7.899747665649382605746887265840, 9.236914866651517190039880379986, 10.18585308697546762253281088239

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