Properties

Label 2-3344-1.1-c1-0-60
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  − 3.21·3-s + 2.30·5-s + 1.34·7-s + 7.31·9-s − 11-s − 1.34·13-s − 7.41·15-s + 0.904·17-s − 19-s − 4.31·21-s − 0.548·23-s + 0.324·25-s − 13.8·27-s − 4.32·29-s − 1.89·31-s + 3.21·33-s + 3.10·35-s − 0.546·37-s + 4.31·39-s − 4.86·41-s − 5.24·43-s + 16.8·45-s + 5.11·47-s − 5.19·49-s − 2.90·51-s − 3.63·53-s − 2.30·55-s + ⋯
L(s)  = 1  − 1.85·3-s + 1.03·5-s + 0.508·7-s + 2.43·9-s − 0.301·11-s − 0.373·13-s − 1.91·15-s + 0.219·17-s − 0.229·19-s − 0.942·21-s − 0.114·23-s + 0.0648·25-s − 2.66·27-s − 0.803·29-s − 0.340·31-s + 0.559·33-s + 0.524·35-s − 0.0898·37-s + 0.691·39-s − 0.759·41-s − 0.799·43-s + 2.51·45-s + 0.745·47-s − 0.741·49-s − 0.406·51-s − 0.499·53-s − 0.311·55-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 \( 1 + 3.21T + 3T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 - 1.34T + 7T^{2} \)
13 \( 1 + 1.34T + 13T^{2} \)
17 \( 1 - 0.904T + 17T^{2} \)
23 \( 1 + 0.548T + 23T^{2} \)
29 \( 1 + 4.32T + 29T^{2} \)
31 \( 1 + 1.89T + 31T^{2} \)
37 \( 1 + 0.546T + 37T^{2} \)
41 \( 1 + 4.86T + 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 - 5.11T + 47T^{2} \)
53 \( 1 + 3.63T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 3.90T + 61T^{2} \)
67 \( 1 - 5.72T + 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 - 1.28T + 73T^{2} \)
79 \( 1 - 6.80T + 79T^{2} \)
83 \( 1 - 7.06T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−8.071414982928944551457707870978, −7.30291018359769037770459087723, −6.47279518295644259323036065949, −5.97524352492748728477794927121, −5.16563313404576267716513931810, −4.89738939191696392152618624346, −3.73743409446342192080737076317, −2.15586229890968112127383941388, −1.36049172356529115273911667401, 0, 1.36049172356529115273911667401, 2.15586229890968112127383941388, 3.73743409446342192080737076317, 4.89738939191696392152618624346, 5.16563313404576267716513931810, 5.97524352492748728477794927121, 6.47279518295644259323036065949, 7.30291018359769037770459087723, 8.071414982928944551457707870978

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