Properties

Label 2-1856-1.1-c3-0-10
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  − 9.22·3-s − 9.91·5-s − 18.1·7-s + 58.0·9-s − 30.3·11-s + 38.5·13-s + 91.4·15-s + 59.0·17-s − 50.7·19-s + 167.·21-s + 155.·23-s − 26.6·25-s − 286.·27-s − 29·29-s − 241.·31-s + 279.·33-s + 179.·35-s − 371.·37-s − 355.·39-s + 61.4·41-s − 369.·43-s − 575.·45-s − 281.·47-s − 13.7·49-s − 545.·51-s + 374.·53-s + 300.·55-s + ⋯
L(s)  = 1  − 1.77·3-s − 0.887·5-s − 0.979·7-s + 2.15·9-s − 0.831·11-s + 0.823·13-s + 1.57·15-s + 0.843·17-s − 0.612·19-s + 1.73·21-s + 1.41·23-s − 0.213·25-s − 2.04·27-s − 0.185·29-s − 1.39·31-s + 1.47·33-s + 0.869·35-s − 1.65·37-s − 1.46·39-s + 0.234·41-s − 1.31·43-s − 1.90·45-s − 0.874·47-s − 0.0400·49-s − 1.49·51-s + 0.969·53-s + 0.737·55-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

\(L(2)\) \(\approx\) \(0.1571094991\)
\(L(\frac12)\) \(\approx\) \(0.1571094991\)
\(L(\frac{5}{2})\) 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 \( 1 + 9.22T + 27T^{2} \)
5 \( 1 + 9.91T + 125T^{2} \)
7 \( 1 + 18.1T + 343T^{2} \)
11 \( 1 + 30.3T + 1.33e3T^{2} \)
13 \( 1 - 38.5T + 2.19e3T^{2} \)
17 \( 1 - 59.0T + 4.91e3T^{2} \)
19 \( 1 + 50.7T + 6.85e3T^{2} \)
23 \( 1 - 155.T + 1.21e4T^{2} \)
31 \( 1 + 241.T + 2.97e4T^{2} \)
37 \( 1 + 371.T + 5.06e4T^{2} \)
41 \( 1 - 61.4T + 6.89e4T^{2} \)
43 \( 1 + 369.T + 7.95e4T^{2} \)
47 \( 1 + 281.T + 1.03e5T^{2} \)
53 \( 1 - 374.T + 1.48e5T^{2} \)
59 \( 1 - 129.T + 2.05e5T^{2} \)
61 \( 1 + 463.T + 2.26e5T^{2} \)
67 \( 1 + 970.T + 3.00e5T^{2} \)
71 \( 1 - 241.T + 3.57e5T^{2} \)
73 \( 1 - 565.T + 3.89e5T^{2} \)
79 \( 1 - 424.T + 4.93e5T^{2} \)
83 \( 1 - 161.T + 5.71e5T^{2} \)
89 \( 1 + 1.13e3T + 7.04e5T^{2} \)
97 \( 1 + 1.44e3T + 9.12e5T^{2} \)
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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

−8.934368969031330432730862355010, −7.88848285671483418885561656616, −7.05595908742856198290422872987, −6.50968436623415758718421361485, −5.58481175058152071930614501315, −5.07848579757850947490342620260, −3.95924242505833978156138775434, −3.22280750525519146151861867644, −1.43431315216554725002984207038, −0.21416710628756728758618672224, 0.21416710628756728758618672224, 1.43431315216554725002984207038, 3.22280750525519146151861867644, 3.95924242505833978156138775434, 5.07848579757850947490342620260, 5.58481175058152071930614501315, 6.50968436623415758718421361485, 7.05595908742856198290422872987, 7.88848285671483418885561656616, 8.934368969031330432730862355010

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