Properties

Label 2-1856-1.1-c3-0-52
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  − 7.82·3-s + 7.04·5-s + 14.1·7-s + 34.2·9-s + 16.9·11-s − 62.0·13-s − 55.1·15-s + 104.·17-s + 9.38·19-s − 110.·21-s + 173.·23-s − 75.4·25-s − 57.1·27-s − 29·29-s − 28.2·31-s − 133.·33-s + 99.6·35-s + 171.·37-s + 485.·39-s + 92.0·41-s + 519.·43-s + 241.·45-s + 109.·47-s − 142.·49-s − 820.·51-s − 140.·53-s + 119.·55-s + ⋯
L(s)  = 1  − 1.50·3-s + 0.629·5-s + 0.763·7-s + 1.27·9-s + 0.465·11-s − 1.32·13-s − 0.949·15-s + 1.49·17-s + 0.113·19-s − 1.15·21-s + 1.57·23-s − 0.603·25-s − 0.407·27-s − 0.185·29-s − 0.163·31-s − 0.702·33-s + 0.481·35-s + 0.762·37-s + 1.99·39-s + 0.350·41-s + 1.84·43-s + 0.800·45-s + 0.340·47-s − 0.416·49-s − 2.25·51-s − 0.364·53-s + 0.293·55-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

\(L(2)\) \(\approx\) \(1.635514246\)
\(L(\frac12)\) \(\approx\) \(1.635514246\)
\(L(\frac{5}{2})\) 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 \( 1 + 7.82T + 27T^{2} \)
5 \( 1 - 7.04T + 125T^{2} \)
7 \( 1 - 14.1T + 343T^{2} \)
11 \( 1 - 16.9T + 1.33e3T^{2} \)
13 \( 1 + 62.0T + 2.19e3T^{2} \)
17 \( 1 - 104.T + 4.91e3T^{2} \)
19 \( 1 - 9.38T + 6.85e3T^{2} \)
23 \( 1 - 173.T + 1.21e4T^{2} \)
31 \( 1 + 28.2T + 2.97e4T^{2} \)
37 \( 1 - 171.T + 5.06e4T^{2} \)
41 \( 1 - 92.0T + 6.89e4T^{2} \)
43 \( 1 - 519.T + 7.95e4T^{2} \)
47 \( 1 - 109.T + 1.03e5T^{2} \)
53 \( 1 + 140.T + 1.48e5T^{2} \)
59 \( 1 + 353.T + 2.05e5T^{2} \)
61 \( 1 - 185.T + 2.26e5T^{2} \)
67 \( 1 + 574.T + 3.00e5T^{2} \)
71 \( 1 - 535.T + 3.57e5T^{2} \)
73 \( 1 + 508.T + 3.89e5T^{2} \)
79 \( 1 + 61.0T + 4.93e5T^{2} \)
83 \( 1 - 826.T + 5.71e5T^{2} \)
89 \( 1 + 938.T + 7.04e5T^{2} \)
97 \( 1 + 888.T + 9.12e5T^{2} \)
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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

−9.161880024289609808234439866647, −7.79677415527092110292813066328, −7.27042560590687566664640331889, −6.30032323580504633682671746924, −5.55229298599754436688436523812, −5.09072890947972885906481939495, −4.26511442615022648066635861166, −2.80769808140121024190190184892, −1.53512241343622635734437426453, −0.69688779632008960473218383317, 0.69688779632008960473218383317, 1.53512241343622635734437426453, 2.80769808140121024190190184892, 4.26511442615022648066635861166, 5.09072890947972885906481939495, 5.55229298599754436688436523812, 6.30032323580504633682671746924, 7.27042560590687566664640331889, 7.79677415527092110292813066328, 9.161880024289609808234439866647

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