Properties

Label 2-3822-1.1-c1-0-19
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 + 1.16·5-s + 6-s − 8-s + 9-s − 1.16·10-s + 1.51·11-s − 12-s + 13-s − 1.16·15-s + 16-s + 5.84·17-s − 18-s − 1.45·19-s + 1.16·20-s − 1.51·22-s + 5.77·23-s + 24-s − 3.65·25-s − 26-s − 27-s + 3.59·29-s + 1.16·30-s + 7.29·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.519·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.366·10-s + 0.458·11-s − 0.288·12-s + 0.277·13-s − 0.299·15-s + 0.250·16-s + 1.41·17-s − 0.235·18-s − 0.334·19-s + 0.259·20-s − 0.323·22-s + 1.20·23-s + 0.204·24-s − 0.730·25-s − 0.196·26-s − 0.192·27-s + 0.668·29-s + 0.211·30-s + 1.31·31-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

\(L(1)\) \(\approx\) \(1.376893456\)
\(L(\frac12)\) \(\approx\) \(1.376893456\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.16T + 5T^{2} \)
11 \( 1 - 1.51T + 11T^{2} \)
17 \( 1 - 5.84T + 17T^{2} \)
19 \( 1 + 1.45T + 19T^{2} \)
23 \( 1 - 5.77T + 23T^{2} \)
29 \( 1 - 3.59T + 29T^{2} \)
31 \( 1 - 7.29T + 31T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 + 9.88T + 41T^{2} \)
43 \( 1 + 1.77T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 9.58T + 53T^{2} \)
59 \( 1 - 0.0156T + 59T^{2} \)
61 \( 1 - 9.20T + 61T^{2} \)
67 \( 1 - 4.07T + 67T^{2} \)
71 \( 1 + 0.358T + 71T^{2} \)
73 \( 1 - 3.53T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 7.33T + 83T^{2} \)
89 \( 1 + 9.20T + 89T^{2} \)
97 \( 1 - 5.19T + 97T^{2} \)
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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

−8.450861750960643812395455658835, −7.901351564080975774961828396109, −6.84447221426147779031714780121, −6.47634375002341755531708346378, −5.59048097936621621343437352663, −4.94670176287732393690577559666, −3.78192660683206969631362869996, −2.85103039270302236708081537727, −1.65951842081416925463445244045, −0.830126892489549106924122889158, 0.830126892489549106924122889158, 1.65951842081416925463445244045, 2.85103039270302236708081537727, 3.78192660683206969631362869996, 4.94670176287732393690577559666, 5.59048097936621621343437352663, 6.47634375002341755531708346378, 6.84447221426147779031714780121, 7.901351564080975774961828396109, 8.450861750960643812395455658835

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