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Properties

Label 2-920-115.68-c1-0-8
Degree $2$
Conductor $920$
Sign $-0.290 - 0.956i$
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  + (1.79 + 1.79i)3-s + (0.878 + 2.05i)5-s + (−1.78 − 1.78i)7-s + 3.46i·9-s + 0.939i·11-s + (2.75 + 2.75i)13-s + (−2.11 + 5.27i)15-s + (−0.360 − 0.360i)17-s + 0.0967·19-s − 6.43i·21-s + (−1.60 + 4.52i)23-s + (−3.45 + 3.61i)25-s + (−0.829 + 0.829i)27-s + 7.97i·29-s + 1.56·31-s + ⋯
L(s)  = 1  + (1.03 + 1.03i)3-s + (0.392 + 0.919i)5-s + (−0.676 − 0.676i)7-s + 1.15i·9-s + 0.283i·11-s + (0.763 + 0.763i)13-s + (−0.546 + 1.36i)15-s + (−0.0874 − 0.0874i)17-s + 0.0221·19-s − 1.40i·21-s + (−0.333 + 0.942i)23-s + (−0.691 + 0.722i)25-s + (−0.159 + 0.159i)27-s + 1.48i·29-s + 0.281·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.290 - 0.956i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ -0.290 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29397 + 1.74565i\)
\(L(\frac12)\) \(\approx\) \(1.29397 + 1.74565i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.878 - 2.05i)T \)
23 \( 1 + (1.60 - 4.52i)T \)
good3 \( 1 + (-1.79 - 1.79i)T + 3iT^{2} \)
7 \( 1 + (1.78 + 1.78i)T + 7iT^{2} \)
11 \( 1 - 0.939iT - 11T^{2} \)
13 \( 1 + (-2.75 - 2.75i)T + 13iT^{2} \)
17 \( 1 + (0.360 + 0.360i)T + 17iT^{2} \)
19 \( 1 - 0.0967T + 19T^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (0.364 + 0.364i)T + 37iT^{2} \)
41 \( 1 + 0.742T + 41T^{2} \)
43 \( 1 + (-4.73 + 4.73i)T - 43iT^{2} \)
47 \( 1 + (-2.67 + 2.67i)T - 47iT^{2} \)
53 \( 1 + (6.26 - 6.26i)T - 53iT^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 + (2.54 + 2.54i)T + 67iT^{2} \)
71 \( 1 + 8.30T + 71T^{2} \)
73 \( 1 + (4.23 + 4.23i)T + 73iT^{2} \)
79 \( 1 - 3.19T + 79T^{2} \)
83 \( 1 + (-6.25 + 6.25i)T - 83iT^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + (-6.66 - 6.66i)T + 97iT^{2} \)
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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

−10.31786069306821300399774333522, −9.407717372090225323356011682971, −9.049769711264420382962037647029, −7.83117430111648973486328324629, −6.96104757781381201491260898348, −6.15982457556572003039450404332, −4.80400806780800051132696020608, −3.64048478369828421936700089402, −3.33616569850807840173832647496, −1.98131572180151344677100824970, 0.937487464454474575292016707094, 2.23732600445867330603242716771, 3.05836106541095213577055563063, 4.36891793253860930340324115151, 5.83072929757125552103253844715, 6.24382696531128548010139294183, 7.52506185951868205970892174207, 8.330823041499256651628325610333, 8.761757108525275261523039105597, 9.542016130038034913443059221664

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