Properties

Label 2-624-1.1-c1-0-0
Degree $2$
Conductor $624$
Sign $1$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2.82·5-s − 2.82·7-s + 9-s + 2·11-s − 13-s + 2.82·15-s + 7.65·17-s + 2.82·19-s + 2.82·21-s + 4·23-s + 3.00·25-s − 27-s + 2·29-s + 1.17·31-s − 2·33-s + 8.00·35-s − 7.65·37-s + 39-s + 5.17·41-s + 1.65·43-s − 2.82·45-s + 11.6·47-s + 1.00·49-s − 7.65·51-s − 2·53-s − 5.65·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.26·5-s − 1.06·7-s + 0.333·9-s + 0.603·11-s − 0.277·13-s + 0.730·15-s + 1.85·17-s + 0.648·19-s + 0.617·21-s + 0.834·23-s + 0.600·25-s − 0.192·27-s + 0.371·29-s + 0.210·31-s − 0.348·33-s + 1.35·35-s − 1.25·37-s + 0.160·39-s + 0.807·41-s + 0.252·43-s − 0.421·45-s + 1.70·47-s + 0.142·49-s − 1.07·51-s − 0.274·53-s − 0.762·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8478840944\)
\(L(\frac12)\) \(\approx\) \(0.8478840944\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 - 5.17T + 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 7.65T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 0.343T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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

−10.62477045879607834404759818228, −9.823494486222086203985184051987, −8.953040170137146208791208275335, −7.72109461726772599695639038795, −7.15809059112063480398815568835, −6.13401519988824419414369840336, −5.10212935991382657628656380930, −3.86923908462354940211166291468, −3.14751495423018683417630639054, −0.828386562033167066759960164945, 0.828386562033167066759960164945, 3.14751495423018683417630639054, 3.86923908462354940211166291468, 5.10212935991382657628656380930, 6.13401519988824419414369840336, 7.15809059112063480398815568835, 7.72109461726772599695639038795, 8.953040170137146208791208275335, 9.823494486222086203985184051987, 10.62477045879607834404759818228

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