Properties

Label 2-3822-1.1-c1-0-14
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 − 3.27·5-s − 6-s − 8-s + 9-s + 3.27·10-s + 5.27·11-s + 12-s + 13-s − 3.27·15-s + 16-s + 7.27·17-s − 18-s − 5.27·19-s − 3.27·20-s − 5.27·22-s − 5.27·23-s − 24-s + 5.72·25-s − 26-s + 27-s + 0.725·29-s + 3.27·30-s − 8·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.46·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.03·10-s + 1.59·11-s + 0.288·12-s + 0.277·13-s − 0.845·15-s + 0.250·16-s + 1.76·17-s − 0.235·18-s − 1.21·19-s − 0.732·20-s − 1.12·22-s − 1.09·23-s − 0.204·24-s + 1.14·25-s − 0.196·26-s + 0.192·27-s + 0.134·29-s + 0.597·30-s − 1.43·31-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: $\chi_{3822} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.350484183\)
\(L(\frac12)\) \(\approx\) \(1.350484183\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.27T + 5T^{2} \)
11 \( 1 - 5.27T + 11T^{2} \)
17 \( 1 - 7.27T + 17T^{2} \)
19 \( 1 + 5.27T + 19T^{2} \)
23 \( 1 + 5.27T + 23T^{2} \)
29 \( 1 - 0.725T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 3.27T + 37T^{2} \)
41 \( 1 - 8.54T + 41T^{2} \)
43 \( 1 - 5.27T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 + 2.54T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.82T + 73T^{2} \)
79 \( 1 + 2.54T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−8.540468440032149254502933065635, −7.68126026754969796509913755469, −7.47420219748482946486561644420, −6.47840322014192927241071559986, −5.72665115840861795775300775652, −4.16009129305030510463352179807, −3.95735505861470362264923657030, −3.07523652984454287831156098822, −1.79978323713822224518898510137, −0.75017728115126000794265505029, 0.75017728115126000794265505029, 1.79978323713822224518898510137, 3.07523652984454287831156098822, 3.95735505861470362264923657030, 4.16009129305030510463352179807, 5.72665115840861795775300775652, 6.47840322014192927241071559986, 7.47420219748482946486561644420, 7.68126026754969796509913755469, 8.540468440032149254502933065635

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