Properties

Label 2-8400-1.1-c1-0-14
Degree $2$
Conductor $8400$
Sign $1$
Analytic cond. $67.0743$
Root an. cond. $8.18989$
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 − 7-s + 9-s + 11-s + 2·13-s − 6·19-s + 21-s + 23-s − 27-s + 29-s + 2·31-s − 33-s + 7·37-s − 2·39-s − 8·41-s − 43-s − 2·47-s + 49-s + 14·53-s + 6·57-s − 10·59-s − 63-s − 3·67-s − 69-s + 9·71-s − 77-s − 79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.37·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s + 0.185·29-s + 0.359·31-s − 0.174·33-s + 1.15·37-s − 0.320·39-s − 1.24·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 0.794·57-s − 1.30·59-s − 0.125·63-s − 0.366·67-s − 0.120·69-s + 1.06·71-s − 0.113·77-s − 0.112·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(67.0743\)
Root analytic conductor: \(8.18989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.362961920\)
\(L(\frac12)\) \(\approx\) \(1.362961920\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−7.77977724282152080708159472828, −6.89054131867419447049138466846, −6.40786214824318910423889284583, −5.87743001296644206035422276886, −5.00165302165832436486674873396, −4.27990370053768247051750985511, −3.61936895414592254677060231704, −2.63910412087330599625429467204, −1.66485489597975843822728301132, −0.59725309347211011904007879845, 0.59725309347211011904007879845, 1.66485489597975843822728301132, 2.63910412087330599625429467204, 3.61936895414592254677060231704, 4.27990370053768247051750985511, 5.00165302165832436486674873396, 5.87743001296644206035422276886, 6.40786214824318910423889284583, 6.89054131867419447049138466846, 7.77977724282152080708159472828

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