Properties

Label 2-44e2-1.1-c1-0-16
Degree $2$
Conductor $1936$
Sign $1$
Analytic cond. $15.4590$
Root an. cond. $3.93179$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 3·5-s − 2·7-s + 6·9-s − 9·15-s + 6·17-s + 4·19-s − 6·21-s − 23-s + 4·25-s + 9·27-s + 8·29-s + 7·31-s + 6·35-s − 37-s − 4·41-s + 6·43-s − 18·45-s + 8·47-s − 3·49-s + 18·51-s + 2·53-s + 12·57-s + 59-s − 4·61-s − 12·63-s + 5·67-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s − 2.32·15-s + 1.45·17-s + 0.917·19-s − 1.30·21-s − 0.208·23-s + 4/5·25-s + 1.73·27-s + 1.48·29-s + 1.25·31-s + 1.01·35-s − 0.164·37-s − 0.624·41-s + 0.914·43-s − 2.68·45-s + 1.16·47-s − 3/7·49-s + 2.52·51-s + 0.274·53-s + 1.58·57-s + 0.130·59-s − 0.512·61-s − 1.51·63-s + 0.610·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1936\)    =    \(2^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(15.4590\)
Root analytic conductor: \(3.93179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1936,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.564371794\)
\(L(\frac12)\) \(\approx\) \(2.564371794\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.979001814119218951761473279764, −8.394339798874914668244789159293, −7.64755444346078841000957674432, −7.34464298147435526282636037125, −6.21679895699676402510107032244, −4.80898200433265196470127405375, −3.87401429056701171587588607766, −3.28681541538821572850087755824, −2.68030351930047444046477575602, −1.04001479698957226761907987971, 1.04001479698957226761907987971, 2.68030351930047444046477575602, 3.28681541538821572850087755824, 3.87401429056701171587588607766, 4.80898200433265196470127405375, 6.21679895699676402510107032244, 7.34464298147435526282636037125, 7.64755444346078841000957674432, 8.394339798874914668244789159293, 8.979001814119218951761473279764

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