Properties

Label 2-1856-1.1-c3-0-137
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.90·3-s + 6.52·5-s + 5.22·7-s − 23.3·9-s + 21.1·11-s − 83.4·13-s + 12.4·15-s + 11.3·17-s + 7.68·19-s + 9.96·21-s + 153.·23-s − 82.3·25-s − 95.9·27-s + 29·29-s + 270.·31-s + 40.2·33-s + 34.1·35-s + 298.·37-s − 159.·39-s − 184.·41-s − 208.·43-s − 152.·45-s − 553.·47-s − 315.·49-s + 21.5·51-s + 321.·53-s + 137.·55-s + ⋯
L(s)  = 1  + 0.366·3-s + 0.583·5-s + 0.282·7-s − 0.865·9-s + 0.579·11-s − 1.78·13-s + 0.214·15-s + 0.161·17-s + 0.0927·19-s + 0.103·21-s + 1.38·23-s − 0.659·25-s − 0.684·27-s + 0.185·29-s + 1.56·31-s + 0.212·33-s + 0.164·35-s + 1.32·37-s − 0.652·39-s − 0.703·41-s − 0.738·43-s − 0.505·45-s − 1.71·47-s − 0.920·49-s + 0.0592·51-s + 0.833·53-s + 0.338·55-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: \(1\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.90T + 27T^{2} \)
5 \( 1 - 6.52T + 125T^{2} \)
7 \( 1 - 5.22T + 343T^{2} \)
11 \( 1 - 21.1T + 1.33e3T^{2} \)
13 \( 1 + 83.4T + 2.19e3T^{2} \)
17 \( 1 - 11.3T + 4.91e3T^{2} \)
19 \( 1 - 7.68T + 6.85e3T^{2} \)
23 \( 1 - 153.T + 1.21e4T^{2} \)
31 \( 1 - 270.T + 2.97e4T^{2} \)
37 \( 1 - 298.T + 5.06e4T^{2} \)
41 \( 1 + 184.T + 6.89e4T^{2} \)
43 \( 1 + 208.T + 7.95e4T^{2} \)
47 \( 1 + 553.T + 1.03e5T^{2} \)
53 \( 1 - 321.T + 1.48e5T^{2} \)
59 \( 1 + 104.T + 2.05e5T^{2} \)
61 \( 1 + 464.T + 2.26e5T^{2} \)
67 \( 1 + 745.T + 3.00e5T^{2} \)
71 \( 1 + 509.T + 3.57e5T^{2} \)
73 \( 1 + 0.374T + 3.89e5T^{2} \)
79 \( 1 - 610.T + 4.93e5T^{2} \)
83 \( 1 + 791.T + 5.71e5T^{2} \)
89 \( 1 + 342.T + 7.04e5T^{2} \)
97 \( 1 - 601.T + 9.12e5T^{2} \)
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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

−8.466387938717131168857234566619, −7.81383669880812795802907851896, −6.90322144254475724542664223182, −6.10858128274664950271612941829, −5.15489385806100733276816374516, −4.53369169869112976909313680943, −3.13394496298783334510622350428, −2.52629396243925500829846400147, −1.41600545151098381940867217950, 0, 1.41600545151098381940867217950, 2.52629396243925500829846400147, 3.13394496298783334510622350428, 4.53369169869112976909313680943, 5.15489385806100733276816374516, 6.10858128274664950271612941829, 6.90322144254475724542664223182, 7.81383669880812795802907851896, 8.466387938717131168857234566619

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