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Properties

Label 2-920-115.68-c1-0-24
Degree $2$
Conductor $920$
Sign $-0.999 - 0.0160i$
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  + (−0.838 − 0.838i)3-s + (−1.37 + 1.76i)5-s + (0.489 + 0.489i)7-s − 1.59i·9-s + 1.81i·11-s + (−0.145 − 0.145i)13-s + (2.62 − 0.331i)15-s + (−2.15 − 2.15i)17-s + 2.11·19-s − 0.820i·21-s + (−4.39 + 1.92i)23-s + (−1.23 − 4.84i)25-s + (−3.85 + 3.85i)27-s − 1.04i·29-s − 8.55·31-s + ⋯
L(s)  = 1  + (−0.483 − 0.483i)3-s + (−0.613 + 0.789i)5-s + (0.185 + 0.185i)7-s − 0.531i·9-s + 0.546i·11-s + (−0.0402 − 0.0402i)13-s + (0.678 − 0.0854i)15-s + (−0.523 − 0.523i)17-s + 0.484·19-s − 0.179i·21-s + (−0.916 + 0.400i)23-s + (−0.247 − 0.968i)25-s + (−0.741 + 0.741i)27-s − 0.193i·29-s − 1.53·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.999 - 0.0160i$
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: \((2,\ 920,\ (\ :1/2),\ -0.999 - 0.0160i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000324145 + 0.0403613i\)
\(L(\frac12)\) \(\approx\) \(0.000324145 + 0.0403613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 - 1.76i)T \)
23 \( 1 + (4.39 - 1.92i)T \)
good3 \( 1 + (0.838 + 0.838i)T + 3iT^{2} \)
7 \( 1 + (-0.489 - 0.489i)T + 7iT^{2} \)
11 \( 1 - 1.81iT - 11T^{2} \)
13 \( 1 + (0.145 + 0.145i)T + 13iT^{2} \)
17 \( 1 + (2.15 + 2.15i)T + 17iT^{2} \)
19 \( 1 - 2.11T + 19T^{2} \)
29 \( 1 + 1.04iT - 29T^{2} \)
31 \( 1 + 8.55T + 31T^{2} \)
37 \( 1 + (4.95 + 4.95i)T + 37iT^{2} \)
41 \( 1 + 1.90T + 41T^{2} \)
43 \( 1 + (0.627 - 0.627i)T - 43iT^{2} \)
47 \( 1 + (3.93 - 3.93i)T - 47iT^{2} \)
53 \( 1 + (4.88 - 4.88i)T - 53iT^{2} \)
59 \( 1 + 0.177iT - 59T^{2} \)
61 \( 1 - 5.66iT - 61T^{2} \)
67 \( 1 + (0.324 + 0.324i)T + 67iT^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + (6.79 + 6.79i)T + 73iT^{2} \)
79 \( 1 - 1.58T + 79T^{2} \)
83 \( 1 + (2.97 - 2.97i)T - 83iT^{2} \)
89 \( 1 - 6.52T + 89T^{2} \)
97 \( 1 + (2.33 + 2.33i)T + 97iT^{2} \)
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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

−9.696229369404346951179885917769, −8.851688047559228637233565295637, −7.63891280221596815915454416728, −7.17852749202016362656109413311, −6.32898308968371981675949281772, −5.41917363691121808940196554445, −4.18274578480953690168810310875, −3.20918721945923059160976427658, −1.85496622544967170721403963757, −0.01969364629498489898510779481, 1.73442285724095439180024246175, 3.47387260807309352597636371838, 4.40359497772856695292653579244, 5.14262821118787964636639464789, 5.98154820471789592298794663432, 7.22766710121455116924082514940, 8.118203340954436575901136194301, 8.718173165886999311369461974091, 9.739054823147560940828631566727, 10.61245337407174269133797474449

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