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Properties

Label 2-920-5.4-c1-0-23
Degree $2$
Conductor $920$
Sign $-0.208 + 0.977i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.40i·3-s + (0.466 − 2.18i)5-s − 1.57i·7-s + 1.03·9-s + 4.35·11-s + 0.964i·13-s + (−3.06 − 0.655i)15-s + 0.300i·17-s + 8.62·19-s − 2.20·21-s − i·23-s + (−4.56 − 2.04i)25-s − 5.65i·27-s − 4.76·29-s − 5.59·31-s + ⋯
L(s)  = 1  − 0.810i·3-s + (0.208 − 0.977i)5-s − 0.593i·7-s + 0.343·9-s + 1.31·11-s + 0.267i·13-s + (−0.792 − 0.169i)15-s + 0.0728i·17-s + 1.97·19-s − 0.481·21-s − 0.208i·23-s + (−0.912 − 0.408i)25-s − 1.08i·27-s − 0.885·29-s − 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.208 + 0.977i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ -0.208 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16001 - 1.43388i\)
\(L(\frac12)\) \(\approx\) \(1.16001 - 1.43388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.466 + 2.18i)T \)
23 \( 1 + iT \)
good3 \( 1 + 1.40iT - 3T^{2} \)
7 \( 1 + 1.57iT - 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
13 \( 1 - 0.964iT - 13T^{2} \)
17 \( 1 - 0.300iT - 17T^{2} \)
19 \( 1 - 8.62T + 19T^{2} \)
29 \( 1 + 4.76T + 29T^{2} \)
31 \( 1 + 5.59T + 31T^{2} \)
37 \( 1 - 4.38iT - 37T^{2} \)
41 \( 1 + 6.62T + 41T^{2} \)
43 \( 1 + 1.72iT - 43T^{2} \)
47 \( 1 - 0.687iT - 47T^{2} \)
53 \( 1 - 8.05iT - 53T^{2} \)
59 \( 1 + 5.74T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 6.49iT - 67T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 6.35iT - 73T^{2} \)
79 \( 1 - 6.95T + 79T^{2} \)
83 \( 1 + 0.185iT - 83T^{2} \)
89 \( 1 - 1.64T + 89T^{2} \)
97 \( 1 - 9.21iT - 97T^{2} \)
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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

−9.505412819231422382856653575322, −9.228249838989303496357252843651, −8.002850965029145863743232473110, −7.31609792046464701545681251376, −6.55109411919356515492130044851, −5.52638102487388432751933904375, −4.47441359635681650101427716652, −3.54970418089357246667871999272, −1.72443413405064397042202014850, −1.01244124474159282961573048731, 1.71038154624222794229191579124, 3.22216692433979513559418067950, 3.80596566494850348908847509667, 5.12003973042275268428475889134, 5.91053216130758548966069325054, 6.97192250087386605156129975957, 7.61289822165355245071308653889, 9.073581247772755032648792438884, 9.476581145005718139902104880332, 10.19048539306740114358482836821

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