Properties

Label 2-3822-1.1-c1-0-41
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 + 3.51·5-s − 6-s + 8-s + 9-s + 3.51·10-s + 3.45·11-s − 12-s − 13-s − 3.51·15-s + 16-s + 1.45·17-s + 18-s + 3.51·19-s + 3.51·20-s + 3.45·22-s + 5.51·23-s − 24-s + 7.34·25-s − 26-s − 27-s − 0.869·29-s − 3.51·30-s − 4.96·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.57·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.11·10-s + 1.04·11-s − 0.288·12-s − 0.277·13-s − 0.907·15-s + 0.250·16-s + 0.352·17-s + 0.235·18-s + 0.805·19-s + 0.785·20-s + 0.736·22-s + 1.14·23-s − 0.204·24-s + 1.46·25-s − 0.196·26-s − 0.192·27-s − 0.161·29-s − 0.641·30-s − 0.892·31-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

\(L(1)\) \(\approx\) \(3.852549495\)
\(L(\frac12)\) \(\approx\) \(3.852549495\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.51T + 5T^{2} \)
11 \( 1 - 3.45T + 11T^{2} \)
17 \( 1 - 1.45T + 17T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
23 \( 1 - 5.51T + 23T^{2} \)
29 \( 1 + 0.869T + 29T^{2} \)
31 \( 1 + 4.96T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 + 3.55T + 41T^{2} \)
43 \( 1 + 4.42T + 43T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 4.96T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 0.486T + 73T^{2} \)
79 \( 1 + 8.62T + 79T^{2} \)
83 \( 1 + 1.36T + 83T^{2} \)
89 \( 1 + 1.33T + 89T^{2} \)
97 \( 1 + 2.08T + 97T^{2} \)
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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

−8.663199962097723984870321530391, −7.35360514713728310802238273222, −6.79720357481497751242211554158, −6.17284725479883470925084804295, −5.34815394825281932592410213184, −5.11278571845715007707861114164, −3.92137392736663518868893100973, −3.01609503226436218307250225305, −1.92726654729030969633296578717, −1.17415510914794452098151642848, 1.17415510914794452098151642848, 1.92726654729030969633296578717, 3.01609503226436218307250225305, 3.92137392736663518868893100973, 5.11278571845715007707861114164, 5.34815394825281932592410213184, 6.17284725479883470925084804295, 6.79720357481497751242211554158, 7.35360514713728310802238273222, 8.663199962097723984870321530391

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