Properties

Label 2-3344-1.1-c1-0-53
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  − 2.11·3-s + 2.11·5-s − 3.11·7-s + 1.47·9-s + 11-s − 1.83·13-s − 4.47·15-s + 4.22·17-s + 19-s + 6.58·21-s − 2.64·23-s − 0.527·25-s + 3.22·27-s + 1.24·29-s − 0.811·31-s − 2.11·33-s − 6.58·35-s + 7.58·37-s + 3.87·39-s − 2.54·41-s − 8.41·43-s + 3.11·45-s + 9.17·47-s + 2.70·49-s − 8.94·51-s − 4.94·53-s + 2.11·55-s + ⋯
L(s)  = 1  − 1.22·3-s + 0.945·5-s − 1.17·7-s + 0.490·9-s + 0.301·11-s − 0.507·13-s − 1.15·15-s + 1.02·17-s + 0.229·19-s + 1.43·21-s − 0.550·23-s − 0.105·25-s + 0.621·27-s + 0.230·29-s − 0.145·31-s − 0.368·33-s − 1.11·35-s + 1.24·37-s + 0.619·39-s − 0.397·41-s − 1.28·43-s + 0.464·45-s + 1.33·47-s + 0.386·49-s − 1.25·51-s − 0.679·53-s + 0.285·55-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 \( 1 + 2.11T + 3T^{2} \)
5 \( 1 - 2.11T + 5T^{2} \)
7 \( 1 + 3.11T + 7T^{2} \)
13 \( 1 + 1.83T + 13T^{2} \)
17 \( 1 - 4.22T + 17T^{2} \)
23 \( 1 + 2.64T + 23T^{2} \)
29 \( 1 - 1.24T + 29T^{2} \)
31 \( 1 + 0.811T + 31T^{2} \)
37 \( 1 - 7.58T + 37T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 + 8.41T + 43T^{2} \)
47 \( 1 - 9.17T + 47T^{2} \)
53 \( 1 + 4.94T + 53T^{2} \)
59 \( 1 + 1.69T + 59T^{2} \)
61 \( 1 + 0.715T + 61T^{2} \)
67 \( 1 - 8.21T + 67T^{2} \)
71 \( 1 + 8.11T + 71T^{2} \)
73 \( 1 + 2.22T + 73T^{2} \)
79 \( 1 - 3.51T + 79T^{2} \)
83 \( 1 + 3.95T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + 1.69T + 97T^{2} \)
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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

−8.240531224370200314784570783396, −7.22387985488260394409944460787, −6.52575724874111789937293189175, −5.91459120260320057526695417012, −5.52664825105120138219851834994, −4.58541389226076648713913708633, −3.47185110883522129634359445653, −2.54034385163488225300900809153, −1.25233195549589881103109462143, 0, 1.25233195549589881103109462143, 2.54034385163488225300900809153, 3.47185110883522129634359445653, 4.58541389226076648713913708633, 5.52664825105120138219851834994, 5.91459120260320057526695417012, 6.52575724874111789937293189175, 7.22387985488260394409944460787, 8.240531224370200314784570783396

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