Properties

Label 2-1856-1.1-c3-0-30
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  + 3.44·3-s − 9.69·5-s − 27.5·7-s − 15.1·9-s + 52.3·11-s + 5.40·13-s − 33.4·15-s + 17.1·17-s − 44.2·19-s − 95.1·21-s − 205.·23-s − 30.9·25-s − 145.·27-s − 29·29-s + 299.·31-s + 180.·33-s + 267.·35-s − 29.7·37-s + 18.6·39-s − 43.9·41-s − 64.8·43-s + 146.·45-s − 499.·47-s + 418.·49-s + 59.3·51-s + 351.·53-s − 507.·55-s + ⋯
L(s)  = 1  + 0.663·3-s − 0.867·5-s − 1.49·7-s − 0.559·9-s + 1.43·11-s + 0.115·13-s − 0.575·15-s + 0.245·17-s − 0.533·19-s − 0.989·21-s − 1.85·23-s − 0.247·25-s − 1.03·27-s − 0.185·29-s + 1.73·31-s + 0.952·33-s + 1.29·35-s − 0.132·37-s + 0.0765·39-s − 0.167·41-s − 0.229·43-s + 0.485·45-s − 1.55·47-s + 1.22·49-s + 0.162·51-s + 0.911·53-s − 1.24·55-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

\(L(2)\) \(\approx\) \(1.154925839\)
\(L(\frac12)\) \(\approx\) \(1.154925839\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.44T + 27T^{2} \)
5 \( 1 + 9.69T + 125T^{2} \)
7 \( 1 + 27.5T + 343T^{2} \)
11 \( 1 - 52.3T + 1.33e3T^{2} \)
13 \( 1 - 5.40T + 2.19e3T^{2} \)
17 \( 1 - 17.1T + 4.91e3T^{2} \)
19 \( 1 + 44.2T + 6.85e3T^{2} \)
23 \( 1 + 205.T + 1.21e4T^{2} \)
31 \( 1 - 299.T + 2.97e4T^{2} \)
37 \( 1 + 29.7T + 5.06e4T^{2} \)
41 \( 1 + 43.9T + 6.89e4T^{2} \)
43 \( 1 + 64.8T + 7.95e4T^{2} \)
47 \( 1 + 499.T + 1.03e5T^{2} \)
53 \( 1 - 351.T + 1.48e5T^{2} \)
59 \( 1 + 522.T + 2.05e5T^{2} \)
61 \( 1 + 484.T + 2.26e5T^{2} \)
67 \( 1 - 504.T + 3.00e5T^{2} \)
71 \( 1 - 481.T + 3.57e5T^{2} \)
73 \( 1 - 3.11T + 3.89e5T^{2} \)
79 \( 1 + 1.04e3T + 4.93e5T^{2} \)
83 \( 1 - 1.00e3T + 5.71e5T^{2} \)
89 \( 1 + 295.T + 7.04e5T^{2} \)
97 \( 1 - 428.T + 9.12e5T^{2} \)
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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

−8.769479737751325647048835913566, −8.244756372905439831809539320694, −7.39741431500271757074391105961, −6.31928024839069932214993420526, −6.10804389688195153774177041288, −4.45592025332053249081284439596, −3.65226383290654908420890471325, −3.22588952422120851375556921165, −2.01156523622587117318376398948, −0.46397687471829912936621347208, 0.46397687471829912936621347208, 2.01156523622587117318376398948, 3.22588952422120851375556921165, 3.65226383290654908420890471325, 4.45592025332053249081284439596, 6.10804389688195153774177041288, 6.31928024839069932214993420526, 7.39741431500271757074391105961, 8.244756372905439831809539320694, 8.769479737751325647048835913566

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