Properties

Label 2-3344-1.1-c1-0-76
Degree $2$
Conductor $3344$
Sign $-1$
Analytic cond. $26.7019$
Root an. cond. $5.16739$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 3·7-s − 2·9-s + 11-s − 13-s − 2·15-s − 7·17-s + 19-s − 3·21-s − 3·23-s − 25-s + 5·27-s − 29-s − 2·31-s − 33-s + 6·35-s − 10·37-s + 39-s − 6·41-s − 8·43-s − 4·45-s + 2·49-s + 7·51-s + 53-s + 2·55-s − 57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.13·7-s − 2/3·9-s + 0.301·11-s − 0.277·13-s − 0.516·15-s − 1.69·17-s + 0.229·19-s − 0.654·21-s − 0.625·23-s − 1/5·25-s + 0.962·27-s − 0.185·29-s − 0.359·31-s − 0.174·33-s + 1.01·35-s − 1.64·37-s + 0.160·39-s − 0.937·41-s − 1.21·43-s − 0.596·45-s + 2/7·49-s + 0.980·51-s + 0.137·53-s + 0.269·55-s − 0.132·57-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: yes
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 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−8.439797524592481347533502047028, −7.45546496090240030764161465905, −6.56091332288600357479592533597, −6.01247644626710348806796859277, −5.10933763800595423955857491965, −4.74637770328277657622171312551, −3.52580991716482041758999285138, −2.22929211602911623327487974279, −1.65462995784748633358443619737, 0, 1.65462995784748633358443619737, 2.22929211602911623327487974279, 3.52580991716482041758999285138, 4.74637770328277657622171312551, 5.10933763800595423955857491965, 6.01247644626710348806796859277, 6.56091332288600357479592533597, 7.45546496090240030764161465905, 8.439797524592481347533502047028

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