Properties

Label 2-3344-1.1-c1-0-84
Degree $2$
Conductor $3344$
Sign $-1$
Analytic cond. $26.7019$
Root an. cond. $5.16739$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s + 2·5-s − 3.56·7-s − 0.561·9-s − 11-s − 3.56·13-s + 3.12·15-s + 3.56·17-s + 19-s − 5.56·21-s − 5.56·23-s − 25-s − 5.56·27-s + 6.68·29-s − 2·31-s − 1.56·33-s − 7.12·35-s + 3.12·37-s − 5.56·39-s + 2·41-s − 1.12·45-s − 8·47-s + 5.68·49-s + 5.56·51-s − 7.80·53-s − 2·55-s + 1.56·57-s + ⋯
L(s)  = 1  + 0.901·3-s + 0.894·5-s − 1.34·7-s − 0.187·9-s − 0.301·11-s − 0.987·13-s + 0.806·15-s + 0.863·17-s + 0.229·19-s − 1.21·21-s − 1.15·23-s − 0.200·25-s − 1.07·27-s + 1.24·29-s − 0.359·31-s − 0.271·33-s − 1.20·35-s + 0.513·37-s − 0.890·39-s + 0.312·41-s − 0.167·45-s − 1.16·47-s + 0.812·49-s + 0.778·51-s − 1.07·53-s − 0.269·55-s + 0.206·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(26.7019\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3344,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 - 1.56T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
13 \( 1 + 3.56T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
23 \( 1 + 5.56T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 3.12T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 7.80T + 53T^{2} \)
59 \( 1 + 4.68T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + 4.68T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 2.68T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 2.24T + 83T^{2} \)
89 \( 1 + 9.12T + 89T^{2} \)
97 \( 1 - 1.12T + 97T^{2} \)
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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

−8.168938202621253209161008188640, −7.70449044677278006478136073664, −6.68660139565452404036237251892, −6.01836922646444643217790656664, −5.37435559285531567292872946628, −4.22657008954923117030858530855, −3.10985526071700341300503710576, −2.78701955004407696067589368274, −1.75470755572988585628446622639, 0, 1.75470755572988585628446622639, 2.78701955004407696067589368274, 3.10985526071700341300503710576, 4.22657008954923117030858530855, 5.37435559285531567292872946628, 6.01836922646444643217790656664, 6.68660139565452404036237251892, 7.70449044677278006478136073664, 8.168938202621253209161008188640

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