Properties

Label 2-3822-1.1-c1-0-20
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 1.34·5-s − 6-s + 8-s + 9-s − 1.34·10-s + 5.25·11-s − 12-s − 13-s + 1.34·15-s + 16-s + 3.25·17-s + 18-s − 1.34·19-s − 1.34·20-s + 5.25·22-s + 0.650·23-s − 24-s − 3.17·25-s − 26-s − 27-s + 0.155·29-s + 1.34·30-s − 1.90·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.603·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.426·10-s + 1.58·11-s − 0.288·12-s − 0.277·13-s + 0.348·15-s + 0.250·16-s + 0.790·17-s + 0.235·18-s − 0.309·19-s − 0.301·20-s + 1.12·22-s + 0.135·23-s − 0.204·24-s − 0.635·25-s − 0.196·26-s − 0.192·27-s + 0.0288·29-s + 0.246·30-s − 0.342·31-s + 0.176·32-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

\(L(1)\) \(\approx\) \(2.418441457\)
\(L(\frac12)\) \(\approx\) \(2.418441457\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 5T^{2} \)
11 \( 1 - 5.25T + 11T^{2} \)
17 \( 1 - 3.25T + 17T^{2} \)
19 \( 1 + 1.34T + 19T^{2} \)
23 \( 1 - 0.650T + 23T^{2} \)
29 \( 1 - 0.155T + 29T^{2} \)
31 \( 1 + 1.90T + 31T^{2} \)
37 \( 1 - 7.59T + 37T^{2} \)
41 \( 1 + 3.32T + 41T^{2} \)
43 \( 1 + 3.16T + 43T^{2} \)
47 \( 1 - 8.15T + 47T^{2} \)
53 \( 1 - 4.49T + 53T^{2} \)
59 \( 1 + 1.90T + 59T^{2} \)
61 \( 1 - 2.26T + 61T^{2} \)
67 \( 1 - 1.88T + 67T^{2} \)
71 \( 1 + 5.30T + 71T^{2} \)
73 \( 1 - 5.34T + 73T^{2} \)
79 \( 1 - 5.74T + 79T^{2} \)
83 \( 1 + 2.95T + 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.368244114003473062916528768270, −7.53037272413282401972613046322, −6.92043670082712182103464478911, −6.18035659699420525960172159841, −5.55664750840795254506148088790, −4.57035822118453322741560789824, −3.99396059435873970067842337852, −3.29443738874146201499626690293, −1.98301042689242168813696265506, −0.861042283911803961196655504857, 0.861042283911803961196655504857, 1.98301042689242168813696265506, 3.29443738874146201499626690293, 3.99396059435873970067842337852, 4.57035822118453322741560789824, 5.55664750840795254506148088790, 6.18035659699420525960172159841, 6.92043670082712182103464478911, 7.53037272413282401972613046322, 8.368244114003473062916528768270

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