Properties

Label 2-1856-1.1-c3-0-114
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  − 1.51·3-s + 6.33·5-s − 1.77·7-s − 24.7·9-s − 48.2·11-s + 6.22·13-s − 9.58·15-s + 34.0·17-s + 54.3·19-s + 2.69·21-s + 189.·23-s − 84.9·25-s + 78.2·27-s + 29·29-s + 37.5·31-s + 73.0·33-s − 11.2·35-s + 61.4·37-s − 9.42·39-s + 164.·41-s + 439.·43-s − 156.·45-s − 165.·47-s − 339.·49-s − 51.4·51-s − 572.·53-s − 305.·55-s + ⋯
L(s)  = 1  − 0.291·3-s + 0.566·5-s − 0.0960·7-s − 0.915·9-s − 1.32·11-s + 0.132·13-s − 0.164·15-s + 0.485·17-s + 0.656·19-s + 0.0279·21-s + 1.71·23-s − 0.679·25-s + 0.557·27-s + 0.185·29-s + 0.217·31-s + 0.385·33-s − 0.0543·35-s + 0.273·37-s − 0.0386·39-s + 0.627·41-s + 1.56·43-s − 0.518·45-s − 0.514·47-s − 0.990·49-s − 0.141·51-s − 1.48·53-s − 0.749·55-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 \( 1 + 1.51T + 27T^{2} \)
5 \( 1 - 6.33T + 125T^{2} \)
7 \( 1 + 1.77T + 343T^{2} \)
11 \( 1 + 48.2T + 1.33e3T^{2} \)
13 \( 1 - 6.22T + 2.19e3T^{2} \)
17 \( 1 - 34.0T + 4.91e3T^{2} \)
19 \( 1 - 54.3T + 6.85e3T^{2} \)
23 \( 1 - 189.T + 1.21e4T^{2} \)
31 \( 1 - 37.5T + 2.97e4T^{2} \)
37 \( 1 - 61.4T + 5.06e4T^{2} \)
41 \( 1 - 164.T + 6.89e4T^{2} \)
43 \( 1 - 439.T + 7.95e4T^{2} \)
47 \( 1 + 165.T + 1.03e5T^{2} \)
53 \( 1 + 572.T + 1.48e5T^{2} \)
59 \( 1 - 543.T + 2.05e5T^{2} \)
61 \( 1 - 811.T + 2.26e5T^{2} \)
67 \( 1 + 759.T + 3.00e5T^{2} \)
71 \( 1 + 973.T + 3.57e5T^{2} \)
73 \( 1 + 814.T + 3.89e5T^{2} \)
79 \( 1 + 379.T + 4.93e5T^{2} \)
83 \( 1 + 87.6T + 5.71e5T^{2} \)
89 \( 1 + 1.49e3T + 7.04e5T^{2} \)
97 \( 1 - 46.6T + 9.12e5T^{2} \)
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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

−8.464690885247981940448444821860, −7.74108526602057969412094832798, −6.89484005969079053259693210169, −5.77587613240125555579217988504, −5.50886809462923766694683423553, −4.56042867851649985015332656422, −3.11902011848613314061275842880, −2.60493701803437284423164880060, −1.19418256649454417993238471225, 0, 1.19418256649454417993238471225, 2.60493701803437284423164880060, 3.11902011848613314061275842880, 4.56042867851649985015332656422, 5.50886809462923766694683423553, 5.77587613240125555579217988504, 6.89484005969079053259693210169, 7.74108526602057969412094832798, 8.464690885247981940448444821860

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