Properties

Label 2-3822-1.1-c1-0-15
Degree $2$
Conductor $3822$
Sign $1$
Analytic cond. $30.5188$
Root an. cond. $5.52438$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 5-s + 6-s − 8-s + 9-s − 10-s + 5·11-s − 12-s − 13-s − 15-s + 16-s − 3·17-s − 18-s − 5·19-s + 20-s − 5·22-s + 9·23-s + 24-s − 4·25-s + 26-s − 27-s − 29-s + 30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.14·19-s + 0.223·20-s − 1.06·22-s + 1.87·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s − 0.185·29-s + 0.182·30-s + 0.359·31-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.250746375\)
\(L(\frac12)\) \(\approx\) \(1.250746375\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.729498038677576237111576315252, −7.69957368229981733623340596987, −6.94019610275138728685401090092, −6.35714465627339102702206525464, −5.81346292482023776850557961057, −4.66246091207002106098334231090, −4.00623879520890965794150539890, −2.71961364944813656746191761526, −1.75377692300691096841277041389, −0.76582271785226971447712249556, 0.76582271785226971447712249556, 1.75377692300691096841277041389, 2.71961364944813656746191761526, 4.00623879520890965794150539890, 4.66246091207002106098334231090, 5.81346292482023776850557961057, 6.35714465627339102702206525464, 6.94019610275138728685401090092, 7.69957368229981733623340596987, 8.729498038677576237111576315252

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