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Properties

Label 2-1920-120.29-c0-0-2
Degree $2$
Conductor $1920$
Sign $1$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
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  − i·3-s + 5-s + 2i·7-s − 9-s − i·15-s + 2·21-s + 25-s + i·27-s + 2·29-s + 2i·35-s − 45-s − 3·49-s − 2i·63-s − i·75-s + 81-s + ⋯
L(s)  = 1  − i·3-s + 5-s + 2i·7-s − 9-s − i·15-s + 2·21-s + 25-s + i·27-s + 2·29-s + 2i·35-s − 45-s − 3·49-s − 2i·63-s − i·75-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337161091\)
\(L(\frac12)\) \(\approx\) \(1.337161091\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 - T \)
good7 \( 1 - 2iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 - T^{2} \)
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

−9.158699775033595978465826499023, −8.658403263914324548882351756993, −8.007443160996929553136465974534, −6.77213597953916518965291958308, −6.21026124085781930824898870166, −5.58909745830828812005455090872, −4.88752387105372933368582222530, −2.97311842504770788806316327359, −2.45206276354445370766216181844, −1.52671242857185594892421627128, 1.11462813380787190878932701799, 2.69067008155724674189853371907, 3.69336988928310007198067120277, 4.49269003201539288075684986284, 5.15471077492020672379834550451, 6.27471380149531546630502330110, 6.88572080223347908746394241417, 7.916606579260750959585291395642, 8.729735099517333899856591114411, 9.722211238201394823824434273095

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