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Properties

Label 2-1617-231.164-c0-0-2
Degree $2$
Conductor $1617$
Sign $0.832 + 0.553i$
Analytic cond. $0.806988$
Root an. cond. $0.898325$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + 0.999·6-s − 8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s − 0.999·15-s + (0.5 + 0.866i)16-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s − 0.999·22-s + (0.500 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + 0.999·6-s − 8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s − 0.999·15-s + (0.5 + 0.866i)16-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s − 0.999·22-s + (0.500 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (1550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8370321091\)
\(L(\frac12)\) \(\approx\) \(0.8370321091\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.821022976267341118193207550928, −8.916284588635784433683883109939, −8.465465375094855332444261748373, −6.71466211089282837924300643830, −6.28264848253337047204634976769, −5.58670645844173138324282845497, −4.30234344894054664396542029958, −3.27810128130799374978327088275, −2.60695190527454019321702115681, −1.04177132096115956346294327111, 1.18512798957665209754617975159, 2.30603342486193238416300307314, 3.87146765116864565511629188137, 5.11074897972509540957663478995, 5.88831428796707506587392675408, 6.51584074870408967413965741912, 7.20289405900183582275832133758, 8.104273149936950236405290100261, 8.610435581770299811252048270738, 9.345213665011043315491438049575

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