Properties

Label 2-3822-1.1-c1-0-11
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 − 0.307·5-s + 6-s − 8-s + 9-s + 0.307·10-s + 1.25·11-s − 12-s + 13-s + 0.307·15-s + 16-s + 2.64·17-s − 18-s + 7.86·19-s − 0.307·20-s − 1.25·22-s − 6.47·23-s + 24-s − 4.90·25-s − 26-s − 27-s + 8.88·29-s − 0.307·30-s − 5.22·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.137·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.0973·10-s + 0.379·11-s − 0.288·12-s + 0.277·13-s + 0.0794·15-s + 0.250·16-s + 0.640·17-s − 0.235·18-s + 1.80·19-s − 0.0688·20-s − 0.268·22-s − 1.35·23-s + 0.204·24-s − 0.981·25-s − 0.196·26-s − 0.192·27-s + 1.64·29-s − 0.0561·30-s − 0.937·31-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

\(L(1)\) \(\approx\) \(1.107207912\)
\(L(\frac12)\) \(\approx\) \(1.107207912\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.307T + 5T^{2} \)
11 \( 1 - 1.25T + 11T^{2} \)
17 \( 1 - 2.64T + 17T^{2} \)
19 \( 1 - 7.86T + 19T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 - 8.88T + 29T^{2} \)
31 \( 1 + 5.22T + 31T^{2} \)
37 \( 1 - 8.44T + 37T^{2} \)
41 \( 1 + 0.192T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 9.74T + 47T^{2} \)
53 \( 1 - 7.74T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 9.38T + 61T^{2} \)
67 \( 1 - 9.62T + 67T^{2} \)
71 \( 1 + 1.56T + 71T^{2} \)
73 \( 1 + 6.83T + 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 - 1.04T + 83T^{2} \)
89 \( 1 + 1.24T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.429873255960323474871716302272, −7.67106396526759761595204212560, −7.27506634509109281592273390108, −6.09941866119697653779583060823, −5.86637308739081627632196487749, −4.75959099094212607849065127461, −3.82813941873210653378196391269, −2.91893768888198608697520250989, −1.65801602236660685962059142644, −0.73173820994508540256734422414, 0.73173820994508540256734422414, 1.65801602236660685962059142644, 2.91893768888198608697520250989, 3.82813941873210653378196391269, 4.75959099094212607849065127461, 5.86637308739081627632196487749, 6.09941866119697653779583060823, 7.27506634509109281592273390108, 7.67106396526759761595204212560, 8.429873255960323474871716302272

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