Properties

Label 2-1856-1.1-c3-0-88
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  + 1.50·3-s − 0.787·5-s + 27.3·7-s − 24.7·9-s + 62.2·11-s + 27.7·13-s − 1.18·15-s + 102.·17-s + 55.9·19-s + 41.0·21-s + 175.·23-s − 124.·25-s − 77.6·27-s + 29·29-s + 199.·31-s + 93.3·33-s − 21.5·35-s − 403.·37-s + 41.7·39-s − 118.·41-s + 354.·43-s + 19.4·45-s − 148.·47-s + 405.·49-s + 153.·51-s − 376.·53-s − 49.0·55-s + ⋯
L(s)  = 1  + 0.288·3-s − 0.0704·5-s + 1.47·7-s − 0.916·9-s + 1.70·11-s + 0.592·13-s − 0.0203·15-s + 1.46·17-s + 0.675·19-s + 0.426·21-s + 1.59·23-s − 0.995·25-s − 0.553·27-s + 0.185·29-s + 1.15·31-s + 0.492·33-s − 0.103·35-s − 1.79·37-s + 0.171·39-s − 0.450·41-s + 1.25·43-s + 0.0645·45-s − 0.461·47-s + 1.18·49-s + 0.422·51-s − 0.974·53-s − 0.120·55-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

\(L(2)\) \(\approx\) \(3.720248555\)
\(L(\frac12)\) \(\approx\) \(3.720248555\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.50T + 27T^{2} \)
5 \( 1 + 0.787T + 125T^{2} \)
7 \( 1 - 27.3T + 343T^{2} \)
11 \( 1 - 62.2T + 1.33e3T^{2} \)
13 \( 1 - 27.7T + 2.19e3T^{2} \)
17 \( 1 - 102.T + 4.91e3T^{2} \)
19 \( 1 - 55.9T + 6.85e3T^{2} \)
23 \( 1 - 175.T + 1.21e4T^{2} \)
31 \( 1 - 199.T + 2.97e4T^{2} \)
37 \( 1 + 403.T + 5.06e4T^{2} \)
41 \( 1 + 118.T + 6.89e4T^{2} \)
43 \( 1 - 354.T + 7.95e4T^{2} \)
47 \( 1 + 148.T + 1.03e5T^{2} \)
53 \( 1 + 376.T + 1.48e5T^{2} \)
59 \( 1 - 329.T + 2.05e5T^{2} \)
61 \( 1 + 789.T + 2.26e5T^{2} \)
67 \( 1 - 147.T + 3.00e5T^{2} \)
71 \( 1 - 254.T + 3.57e5T^{2} \)
73 \( 1 + 402.T + 3.89e5T^{2} \)
79 \( 1 + 429.T + 4.93e5T^{2} \)
83 \( 1 - 78.8T + 5.71e5T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 - 1.79e3T + 9.12e5T^{2} \)
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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

−8.842090568529986385629521864809, −8.146966292237341166845884759076, −7.50346307979362754661114171753, −6.48254263695665383066467431049, −5.58738240081658730780004479221, −4.86056805040080265365912820547, −3.79467926194603902812240423912, −3.05654415696120609907088996755, −1.63103005128218280585702751576, −1.01150362163048291681194301486, 1.01150362163048291681194301486, 1.63103005128218280585702751576, 3.05654415696120609907088996755, 3.79467926194603902812240423912, 4.86056805040080265365912820547, 5.58738240081658730780004479221, 6.48254263695665383066467431049, 7.50346307979362754661114171753, 8.146966292237341166845884759076, 8.842090568529986385629521864809

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