Properties

Label 2-1856-1.1-c3-0-7
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.24·3-s − 5.77·5-s + 10.3·7-s + 12.0·9-s − 21.9·11-s − 61.7·13-s + 36.0·15-s + 41.9·17-s − 141.·19-s − 64.4·21-s − 193.·23-s − 91.6·25-s + 93.6·27-s − 29·29-s + 115.·31-s + 137.·33-s − 59.6·35-s + 99.1·37-s + 385.·39-s + 244.·41-s + 385.·43-s − 69.3·45-s − 108.·47-s − 236.·49-s − 261.·51-s − 515.·53-s + 126.·55-s + ⋯
L(s)  = 1  − 1.20·3-s − 0.516·5-s + 0.557·7-s + 0.444·9-s − 0.601·11-s − 1.31·13-s + 0.621·15-s + 0.598·17-s − 1.71·19-s − 0.670·21-s − 1.75·23-s − 0.732·25-s + 0.667·27-s − 0.185·29-s + 0.667·31-s + 0.723·33-s − 0.288·35-s + 0.440·37-s + 1.58·39-s + 0.931·41-s + 1.36·43-s − 0.229·45-s − 0.336·47-s − 0.689·49-s − 0.718·51-s − 1.33·53-s + 0.310·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1660702902\)
\(L(\frac12)\) \(\approx\) \(0.1660702902\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + 29T \)
good3 \( 1 + 6.24T + 27T^{2} \)
5 \( 1 + 5.77T + 125T^{2} \)
7 \( 1 - 10.3T + 343T^{2} \)
11 \( 1 + 21.9T + 1.33e3T^{2} \)
13 \( 1 + 61.7T + 2.19e3T^{2} \)
17 \( 1 - 41.9T + 4.91e3T^{2} \)
19 \( 1 + 141.T + 6.85e3T^{2} \)
23 \( 1 + 193.T + 1.21e4T^{2} \)
31 \( 1 - 115.T + 2.97e4T^{2} \)
37 \( 1 - 99.1T + 5.06e4T^{2} \)
41 \( 1 - 244.T + 6.89e4T^{2} \)
43 \( 1 - 385.T + 7.95e4T^{2} \)
47 \( 1 + 108.T + 1.03e5T^{2} \)
53 \( 1 + 515.T + 1.48e5T^{2} \)
59 \( 1 + 856.T + 2.05e5T^{2} \)
61 \( 1 + 402.T + 2.26e5T^{2} \)
67 \( 1 + 758.T + 3.00e5T^{2} \)
71 \( 1 + 696.T + 3.57e5T^{2} \)
73 \( 1 + 1.10e3T + 3.89e5T^{2} \)
79 \( 1 + 817.T + 4.93e5T^{2} \)
83 \( 1 + 1.08e3T + 5.71e5T^{2} \)
89 \( 1 - 119.T + 7.04e5T^{2} \)
97 \( 1 - 1.60e3T + 9.12e5T^{2} \)
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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

−8.802562065003615790623467336774, −7.77343216708864775250292176129, −7.55615151681721210480626859523, −6.13035523799313724968940764892, −5.89192229532633671314963214403, −4.59777969703257899042473828142, −4.42833554525674082943831469109, −2.84255941221592549156285613666, −1.75299680043428732461771447213, −0.19542659623613621642595570859, 0.19542659623613621642595570859, 1.75299680043428732461771447213, 2.84255941221592549156285613666, 4.42833554525674082943831469109, 4.59777969703257899042473828142, 5.89192229532633671314963214403, 6.13035523799313724968940764892, 7.55615151681721210480626859523, 7.77343216708864775250292176129, 8.802562065003615790623467336774

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