Properties

Label 2-1856-1.1-c3-0-33
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  + 0.757·3-s + 10.6·5-s − 22.1·7-s − 26.4·9-s − 39.3·11-s − 23.7·13-s + 8.07·15-s + 4.54·17-s + 155.·19-s − 16.7·21-s − 41.8·23-s − 11.4·25-s − 40.4·27-s − 29·29-s − 57.9·31-s − 29.7·33-s − 235.·35-s − 235.·37-s − 18.0·39-s − 175.·41-s + 402.·43-s − 281.·45-s + 227.·47-s + 147.·49-s + 3.44·51-s − 673.·53-s − 419.·55-s + ⋯
L(s)  = 1  + 0.145·3-s + 0.953·5-s − 1.19·7-s − 0.978·9-s − 1.07·11-s − 0.507·13-s + 0.138·15-s + 0.0648·17-s + 1.87·19-s − 0.174·21-s − 0.379·23-s − 0.0914·25-s − 0.288·27-s − 0.185·29-s − 0.335·31-s − 0.157·33-s − 1.13·35-s − 1.04·37-s − 0.0739·39-s − 0.667·41-s + 1.42·43-s − 0.932·45-s + 0.706·47-s + 0.429·49-s + 0.00944·51-s − 1.74·53-s − 1.02·55-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

\(L(2)\) \(\approx\) \(1.466373694\)
\(L(\frac12)\) \(\approx\) \(1.466373694\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.757T + 27T^{2} \)
5 \( 1 - 10.6T + 125T^{2} \)
7 \( 1 + 22.1T + 343T^{2} \)
11 \( 1 + 39.3T + 1.33e3T^{2} \)
13 \( 1 + 23.7T + 2.19e3T^{2} \)
17 \( 1 - 4.54T + 4.91e3T^{2} \)
19 \( 1 - 155.T + 6.85e3T^{2} \)
23 \( 1 + 41.8T + 1.21e4T^{2} \)
31 \( 1 + 57.9T + 2.97e4T^{2} \)
37 \( 1 + 235.T + 5.06e4T^{2} \)
41 \( 1 + 175.T + 6.89e4T^{2} \)
43 \( 1 - 402.T + 7.95e4T^{2} \)
47 \( 1 - 227.T + 1.03e5T^{2} \)
53 \( 1 + 673.T + 1.48e5T^{2} \)
59 \( 1 - 800.T + 2.05e5T^{2} \)
61 \( 1 - 222.T + 2.26e5T^{2} \)
67 \( 1 - 524.T + 3.00e5T^{2} \)
71 \( 1 + 281.T + 3.57e5T^{2} \)
73 \( 1 - 1.22e3T + 3.89e5T^{2} \)
79 \( 1 - 611.T + 4.93e5T^{2} \)
83 \( 1 + 515.T + 5.71e5T^{2} \)
89 \( 1 + 358.T + 7.04e5T^{2} \)
97 \( 1 - 829.T + 9.12e5T^{2} \)
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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

−9.111354167251516207551521307130, −8.074222267481988160080804991160, −7.31687091415235579985720472760, −6.39841819151837979958539482796, −5.53498564000533182793210320369, −5.23181109589638870392600909325, −3.58856983184278774633463402859, −2.88202057417799768649202659400, −2.10526997096421772415782900806, −0.52912452322119245020990641007, 0.52912452322119245020990641007, 2.10526997096421772415782900806, 2.88202057417799768649202659400, 3.58856983184278774633463402859, 5.23181109589638870392600909325, 5.53498564000533182793210320369, 6.39841819151837979958539482796, 7.31687091415235579985720472760, 8.074222267481988160080804991160, 9.111354167251516207551521307130

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