Properties

Label 2-8112-1.1-c1-0-4
Degree $2$
Conductor $8112$
Sign $1$
Analytic cond. $64.7746$
Root an. cond. $8.04826$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 0.561·5-s − 3.56·7-s + 9-s − 2·11-s + 0.561·15-s + 2.56·17-s − 1.12·19-s + 3.56·21-s − 2·23-s − 4.68·25-s − 27-s − 5.68·29-s − 1.56·31-s + 2·33-s + 2·35-s − 3.43·37-s − 2.56·41-s − 0.438·43-s − 0.561·45-s − 8.24·47-s + 5.68·49-s − 2.56·51-s + 11.6·53-s + 1.12·55-s + 1.12·57-s − 11.1·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.251·5-s − 1.34·7-s + 0.333·9-s − 0.603·11-s + 0.144·15-s + 0.621·17-s − 0.257·19-s + 0.777·21-s − 0.417·23-s − 0.936·25-s − 0.192·27-s − 1.05·29-s − 0.280·31-s + 0.348·33-s + 0.338·35-s − 0.565·37-s − 0.400·41-s − 0.0668·43-s − 0.0837·45-s − 1.20·47-s + 0.812·49-s − 0.358·51-s + 1.60·53-s + 0.151·55-s + 0.148·57-s − 1.44·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8112\)    =    \(2^{4} \cdot 3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(64.7746\)
Root analytic conductor: \(8.04826\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8112,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4855054791\)
\(L(\frac12)\) \(\approx\) \(0.4855054791\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
13 \( 1 \)
good5 \( 1 + 0.561T + 5T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 - 2.56T + 17T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 + 2.56T + 41T^{2} \)
43 \( 1 + 0.438T + 43T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 0.438T + 67T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 - 1.87T + 73T^{2} \)
79 \( 1 + 9.56T + 79T^{2} \)
83 \( 1 + 9.12T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 4.43T + 97T^{2} \)
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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

−7.75989882426530774426488202845, −7.00574725072672646084387518420, −6.49111584323915903448006788378, −5.65127646323977032447569880297, −5.30857740717724732251103929269, −4.12486866170617628799417483280, −3.60434585419309175848287516954, −2.76627999624588944411643042959, −1.72440110865859184768852377500, −0.34215454935970406850878743825, 0.34215454935970406850878743825, 1.72440110865859184768852377500, 2.76627999624588944411643042959, 3.60434585419309175848287516954, 4.12486866170617628799417483280, 5.30857740717724732251103929269, 5.65127646323977032447569880297, 6.49111584323915903448006788378, 7.00574725072672646084387518420, 7.75989882426530774426488202845

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