Properties

Label 2-3822-1.1-c1-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 2.61·5-s − 6-s − 8-s + 9-s + 2.61·10-s − 3.55·11-s + 12-s − 13-s − 2.61·15-s + 16-s + 7.95·17-s − 18-s + 2.17·19-s − 2.61·20-s + 3.55·22-s − 2.55·23-s − 24-s + 1.82·25-s + 26-s + 27-s − 3·29-s + 2.61·30-s + 0.828·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.16·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.826·10-s − 1.07·11-s + 0.288·12-s − 0.277·13-s − 0.674·15-s + 0.250·16-s + 1.92·17-s − 0.235·18-s + 0.498·19-s − 0.584·20-s + 0.758·22-s − 0.533·23-s − 0.204·24-s + 0.365·25-s + 0.196·26-s + 0.192·27-s − 0.557·29-s + 0.477·30-s + 0.148·31-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: \(1\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.61T + 5T^{2} \)
11 \( 1 + 3.55T + 11T^{2} \)
17 \( 1 - 7.95T + 17T^{2} \)
19 \( 1 - 2.17T + 19T^{2} \)
23 \( 1 + 2.55T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 0.828T + 31T^{2} \)
37 \( 1 - 7.78T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 + 0.773T + 43T^{2} \)
47 \( 1 - 2.94T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 4.78T + 59T^{2} \)
61 \( 1 + 6.72T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 + 7.82T + 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + 2.39T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 4.21T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−8.130858904056281548866533661730, −7.51328561520881343456728042037, −7.21474648018584727348480981586, −5.89331222566998407995159637505, −5.16102648122241101840843222641, −4.05773382380719118038019486489, −3.29947709362510016722678506907, −2.58410224238832967050439894044, −1.28003380186632762188124975126, 0, 1.28003380186632762188124975126, 2.58410224238832967050439894044, 3.29947709362510016722678506907, 4.05773382380719118038019486489, 5.16102648122241101840843222641, 5.89331222566998407995159637505, 7.21474648018584727348480981586, 7.51328561520881343456728042037, 8.130858904056281548866533661730

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