Properties

Label 2-3822-1.1-c1-0-34
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 + 4.27·5-s − 6-s − 8-s + 9-s − 4.27·10-s − 2.27·11-s + 12-s + 13-s + 4.27·15-s + 16-s − 0.274·17-s − 18-s + 2.27·19-s + 4.27·20-s + 2.27·22-s + 2.27·23-s − 24-s + 13.2·25-s − 26-s + 27-s + 8.27·29-s − 4.27·30-s − 8·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.91·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.35·10-s − 0.685·11-s + 0.288·12-s + 0.277·13-s + 1.10·15-s + 0.250·16-s − 0.0666·17-s − 0.235·18-s + 0.521·19-s + 0.955·20-s + 0.485·22-s + 0.474·23-s − 0.204·24-s + 2.65·25-s − 0.196·26-s + 0.192·27-s + 1.53·29-s − 0.780·30-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

\(L(1)\) \(\approx\) \(2.590456619\)
\(L(\frac12)\) \(\approx\) \(2.590456619\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.27T + 5T^{2} \)
11 \( 1 + 2.27T + 11T^{2} \)
17 \( 1 + 0.274T + 17T^{2} \)
19 \( 1 - 2.27T + 19T^{2} \)
23 \( 1 - 2.27T + 23T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 4.27T + 37T^{2} \)
41 \( 1 + 6.54T + 41T^{2} \)
43 \( 1 + 2.27T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 4.54T + 83T^{2} \)
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

−8.632718934712415615076700767581, −7.974300782950850933387185747782, −6.88517958231511771567876718046, −6.54710810803300730587817640424, −5.42521124514534574699953283675, −5.10741852580786120667089600485, −3.53998009425476247684296362885, −2.62025861886281616824784174913, −2.01144168489934844065351739579, −1.06710519062030289478825316279, 1.06710519062030289478825316279, 2.01144168489934844065351739579, 2.62025861886281616824784174913, 3.53998009425476247684296362885, 5.10741852580786120667089600485, 5.42521124514534574699953283675, 6.54710810803300730587817640424, 6.88517958231511771567876718046, 7.974300782950850933387185747782, 8.632718934712415615076700767581

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