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Properties

Label 2-1020-1020.419-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6669966710\)
\(L(\frac12)\) \(\approx\) \(0.6669966710\)
\(L(1)\) 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

−10.19950393601395368505164665916, −9.394510952835769548459446119723, −8.956185594880236563677767216128, −7.35013438057075793026775349158, −6.95221585229642849355105804100, −6.20140673609797837889003104114, −5.17251923733840420285089898080, −4.79370030419440285852745799904, −2.91758175097654918494215683547, −1.19101894754925677556364277854, 1.15961611182617140704787241528, 2.12957255821904708995357952197, 3.72206394089092803600865139034, 4.76730168561373197031008753282, 5.67225928656602656876738153465, 6.54336751330411878569875470658, 7.68378581287742288557841831815, 8.448456151060514160997115311067, 9.513702301090795110220739475556, 10.02538242519826405192750026797

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