Properties

Label 2-3344-1.1-c1-0-4
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.223·3-s − 4.30·5-s + 0.878·7-s − 2.94·9-s − 11-s − 4.33·13-s − 0.963·15-s − 5.89·17-s + 19-s + 0.196·21-s − 4.92·23-s + 13.5·25-s − 1.33·27-s + 3.35·29-s − 2.82·31-s − 0.223·33-s − 3.78·35-s − 1.07·37-s − 0.970·39-s − 0.852·41-s + 7.50·43-s + 12.6·45-s − 12.6·47-s − 6.22·49-s − 1.31·51-s + 13.1·53-s + 4.30·55-s + ⋯
L(s)  = 1  + 0.129·3-s − 1.92·5-s + 0.332·7-s − 0.983·9-s − 0.301·11-s − 1.20·13-s − 0.248·15-s − 1.42·17-s + 0.229·19-s + 0.0428·21-s − 1.02·23-s + 2.70·25-s − 0.256·27-s + 0.623·29-s − 0.507·31-s − 0.0389·33-s − 0.638·35-s − 0.176·37-s − 0.155·39-s − 0.133·41-s + 1.14·43-s + 1.89·45-s − 1.83·47-s − 0.889·49-s − 0.184·51-s + 1.81·53-s + 0.580·55-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: \(0\)
Selberg data: \((2,\ 3344,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4690374890\)
\(L(\frac12)\) \(\approx\) \(0.4690374890\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.223T + 3T^{2} \)
5 \( 1 + 4.30T + 5T^{2} \)
7 \( 1 - 0.878T + 7T^{2} \)
13 \( 1 + 4.33T + 13T^{2} \)
17 \( 1 + 5.89T + 17T^{2} \)
23 \( 1 + 4.92T + 23T^{2} \)
29 \( 1 - 3.35T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + 1.07T + 37T^{2} \)
41 \( 1 + 0.852T + 41T^{2} \)
43 \( 1 - 7.50T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 9.06T + 59T^{2} \)
61 \( 1 + 4.02T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 3.39T + 71T^{2} \)
73 \( 1 + 4.50T + 73T^{2} \)
79 \( 1 - 1.31T + 79T^{2} \)
83 \( 1 + 9.71T + 83T^{2} \)
89 \( 1 - 0.835T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−8.395202390607386863539326090366, −8.004490808612990418738791505313, −7.27320968824090793928620264229, −6.62191680461329878948050546717, −5.39753893246883101981589241386, −4.64777228487949803991970770962, −4.01120033272863343910426842725, −3.08234979131084153970396068417, −2.25173688249219537138368524004, −0.37834104367879552518582411484, 0.37834104367879552518582411484, 2.25173688249219537138368524004, 3.08234979131084153970396068417, 4.01120033272863343910426842725, 4.64777228487949803991970770962, 5.39753893246883101981589241386, 6.62191680461329878948050546717, 7.27320968824090793928620264229, 8.004490808612990418738791505313, 8.395202390607386863539326090366

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