Properties

Label 2-1856-1.1-c1-0-22
Degree $2$
Conductor $1856$
Sign $-1$
Analytic cond. $14.8202$
Root an. cond. $3.84970$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.36·3-s − 4.14·5-s − 1.14·9-s + 5.64·11-s + 2.86·13-s + 5.64·15-s + 2·17-s − 4.28·19-s − 2.72·23-s + 12.1·25-s + 5.64·27-s − 29-s − 5.36·31-s − 7.69·33-s + 6.28·37-s − 3.91·39-s + 11.7·41-s − 2.91·43-s + 4.72·45-s − 4.19·47-s − 7·49-s − 2.72·51-s − 1.41·53-s − 23.3·55-s + 5.83·57-s − 1.27·59-s − 3.45·61-s + ⋯
L(s)  = 1  − 0.787·3-s − 1.85·5-s − 0.380·9-s + 1.70·11-s + 0.795·13-s + 1.45·15-s + 0.485·17-s − 0.982·19-s − 0.568·23-s + 2.43·25-s + 1.08·27-s − 0.185·29-s − 0.963·31-s − 1.33·33-s + 1.03·37-s − 0.626·39-s + 1.83·41-s − 0.445·43-s + 0.704·45-s − 0.611·47-s − 49-s − 0.381·51-s − 0.194·53-s − 3.15·55-s + 0.773·57-s − 0.165·59-s − 0.442·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $-1$
Analytic conductor: \(14.8202\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1856,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + T \)
good3 \( 1 + 1.36T + 3T^{2} \)
5 \( 1 + 4.14T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 - 2.86T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4.28T + 19T^{2} \)
23 \( 1 + 2.72T + 23T^{2} \)
31 \( 1 + 5.36T + 31T^{2} \)
37 \( 1 - 6.28T + 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 + 4.19T + 47T^{2} \)
53 \( 1 + 1.41T + 53T^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 + 3.45T + 61T^{2} \)
67 \( 1 + 9.45T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 7.73T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 9.27T + 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−8.661277786226985694462494277155, −8.140692090360673915117366412993, −7.22119742503973344024085547403, −6.44109397423777392220804178542, −5.77988100972354318878948893235, −4.43970306943864635980738033451, −4.02249764325837518532076433960, −3.13454810722372892081693506686, −1.24264055230817151923655035775, 0, 1.24264055230817151923655035775, 3.13454810722372892081693506686, 4.02249764325837518532076433960, 4.43970306943864635980738033451, 5.77988100972354318878948893235, 6.44109397423777392220804178542, 7.22119742503973344024085547403, 8.140692090360673915117366412993, 8.661277786226985694462494277155

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