Properties

Label 2-38640-1.1-c1-0-27
Degree $2$
Conductor $38640$
Sign $1$
Analytic cond. $308.541$
Root an. cond. $17.5653$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s + 9-s + 4·11-s + 2·13-s + 15-s + 6·17-s + 4·19-s + 21-s − 23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·33-s + 35-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s + 49-s − 6·51-s − 10·53-s − 4·55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 1.37·53-s − 0.539·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(308.541\)
Root analytic conductor: \(17.5653\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 38640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.359153245\)
\(L(\frac12)\) \(\approx\) \(2.359153245\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.64950678484152, −14.32923022645906, −13.97920036200965, −13.09618169227274, −12.62124658914550, −12.23428403155314, −11.58190535742317, −11.35210422280178, −10.71333775810350, −9.988643503867096, −9.574564398645474, −9.132377574348403, −8.299272905575972, −7.832570185681552, −7.217778534091722, −6.607194417439247, −6.149557267540288, −5.518535616811076, −4.956371249544373, −4.149079766412096, −3.569257932182414, −3.208391938822395, −2.078895582612064, −1.120378364569203, −0.7185259607642632, 0.7185259607642632, 1.120378364569203, 2.078895582612064, 3.208391938822395, 3.569257932182414, 4.149079766412096, 4.956371249544373, 5.518535616811076, 6.149557267540288, 6.607194417439247, 7.217778534091722, 7.832570185681552, 8.299272905575972, 9.132377574348403, 9.574564398645474, 9.988643503867096, 10.71333775810350, 11.35210422280178, 11.58190535742317, 12.23428403155314, 12.62124658914550, 13.09618169227274, 13.97920036200965, 14.32923022645906, 14.64950678484152

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