Properties

Label 2-1856-1.1-c3-0-58
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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.772625872\)
\(L(\frac12)\) \(\approx\) \(2.772625872\)
\(L(\frac{5}{2})\) 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

−9.061982548476582115411654538240, −7.958714754970774779357869749634, −7.50527849664156762954694932762, −6.81784403029264906535375802363, −5.56908846584782350903072362183, −4.70510294221324995183747972860, −3.42686204839562239229608609529, −3.21983100560939985713458780442, −2.07271723782660384044000554648, −0.72430777680275883827450942808, 0.72430777680275883827450942808, 2.07271723782660384044000554648, 3.21983100560939985713458780442, 3.42686204839562239229608609529, 4.70510294221324995183747972860, 5.56908846584782350903072362183, 6.81784403029264906535375802363, 7.50527849664156762954694932762, 7.958714754970774779357869749634, 9.061982548476582115411654538240

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