Properties

Label 2-4608-8.5-c1-0-27
Degree $2$
Conductor $4608$
Sign $1$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.08i·5-s − 5.03·7-s − 0.828i·11-s − 2.94i·13-s − 4.82·17-s + 2.82i·19-s − 4.16·23-s + 0.656·25-s − 7.97i·29-s − 5.03·31-s − 10.4i·35-s + 7.11i·37-s + 8.82·41-s + 12.4i·43-s + 4.16·47-s + ⋯
L(s)  = 1  + 0.932i·5-s − 1.90·7-s − 0.249i·11-s − 0.817i·13-s − 1.17·17-s + 0.648i·19-s − 0.869·23-s + 0.131·25-s − 1.48i·29-s − 0.903·31-s − 1.77i·35-s + 1.16i·37-s + 1.37·41-s + 1.90i·43-s + 0.607·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9175234209\)
\(L(\frac12)\) \(\approx\) \(0.9175234209\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 2.08iT - 5T^{2} \)
7 \( 1 + 5.03T + 7T^{2} \)
11 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + 2.94iT - 13T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + 4.16T + 23T^{2} \)
29 \( 1 + 7.97iT - 29T^{2} \)
31 \( 1 + 5.03T + 31T^{2} \)
37 \( 1 - 7.11iT - 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 - 12.4iT - 43T^{2} \)
47 \( 1 - 4.16T + 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + 1.65iT - 59T^{2} \)
61 \( 1 + 7.11iT - 61T^{2} \)
67 \( 1 + 2.34iT - 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 5.03T + 79T^{2} \)
83 \( 1 + 3.17iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.213052772614479323489728757309, −7.54631485655975726947662223104, −6.63988014366753708683793329900, −6.28165114063932994727881201954, −5.72659621995582539921082742105, −4.40705363982815864757334252496, −3.54970026172621352009431139865, −2.99266904312020827319797213305, −2.22889121398405526136876112672, −0.44469414081934315029311288120, 0.57044281098716144365341057894, 1.98081251964077312322059457776, 2.88117916471366819731857548091, 3.93961769258695335163855434281, 4.40156212612306335432423895407, 5.51065545099226300603377625546, 6.09632481637188368336992444026, 7.05766763598886458734889342040, 7.22417458320238813252676039729, 8.714126009586454381326180467165

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