Properties

Label 2-3822-1.1-c1-0-58
Degree $2$
Conductor $3822$
Sign $1$
Analytic cond. $30.5188$
Root an. cond. $5.52438$
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  + 2-s + 3-s + 4-s + 3.70·5-s + 6-s + 8-s + 9-s + 3.70·10-s + 5.70·11-s + 12-s − 13-s + 3.70·15-s + 16-s − 3.70·17-s + 18-s − 5.70·19-s + 3.70·20-s + 5.70·22-s − 1.70·23-s + 24-s + 8.70·25-s − 26-s + 27-s − 3.70·29-s + 3.70·30-s + 32-s + 5.70·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.65·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.17·10-s + 1.71·11-s + 0.288·12-s − 0.277·13-s + 0.955·15-s + 0.250·16-s − 0.897·17-s + 0.235·18-s − 1.30·19-s + 0.827·20-s + 1.21·22-s − 0.354·23-s + 0.204·24-s + 1.74·25-s − 0.196·26-s + 0.192·27-s − 0.687·29-s + 0.675·30-s + 0.176·32-s + 0.992·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3822\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(30.5188\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.500330280\)
\(L(\frac12)\) \(\approx\) \(5.500330280\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
13 \( 1 + T \)
good5 \( 1 - 3.70T + 5T^{2} \)
11 \( 1 - 5.70T + 11T^{2} \)
17 \( 1 + 3.70T + 17T^{2} \)
19 \( 1 + 5.70T + 19T^{2} \)
23 \( 1 + 1.70T + 23T^{2} \)
29 \( 1 + 3.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4.29T + 37T^{2} \)
41 \( 1 - 9.40T + 41T^{2} \)
43 \( 1 - 9.10T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 7.70T + 61T^{2} \)
67 \( 1 + 7.40T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 + 3.40T + 79T^{2} \)
83 \( 1 - 0.596T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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

−8.792089130817399362758465445364, −7.59584046041812661538650921092, −6.75608033364335078939690047310, −6.16744775757243342477793843212, −5.74292246227952023492002227871, −4.42456190823828619421873725895, −4.12674046533058564668665622287, −2.80037503134609174492594909662, −2.12843806786283010744424863656, −1.38638969543504239388096097368, 1.38638969543504239388096097368, 2.12843806786283010744424863656, 2.80037503134609174492594909662, 4.12674046533058564668665622287, 4.42456190823828619421873725895, 5.74292246227952023492002227871, 6.16744775757243342477793843212, 6.75608033364335078939690047310, 7.59584046041812661538650921092, 8.792089130817399362758465445364

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