Properties

Label 2-3344-1.1-c1-0-48
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  − 2.26·3-s + 0.637·5-s − 2.66·7-s + 2.12·9-s − 11-s − 1.20·13-s − 1.44·15-s + 0.222·17-s + 19-s + 6.04·21-s + 9.48·23-s − 4.59·25-s + 1.97·27-s + 4.94·29-s + 3.05·31-s + 2.26·33-s − 1.70·35-s + 7.18·37-s + 2.73·39-s − 6.92·41-s − 1.53·43-s + 1.35·45-s − 5.94·47-s + 0.125·49-s − 0.502·51-s − 9.63·53-s − 0.637·55-s + ⋯
L(s)  = 1  − 1.30·3-s + 0.285·5-s − 1.00·7-s + 0.709·9-s − 0.301·11-s − 0.334·13-s − 0.372·15-s + 0.0538·17-s + 0.229·19-s + 1.31·21-s + 1.97·23-s − 0.918·25-s + 0.380·27-s + 0.918·29-s + 0.548·31-s + 0.394·33-s − 0.287·35-s + 1.18·37-s + 0.437·39-s − 1.08·41-s − 0.234·43-s + 0.202·45-s − 0.867·47-s + 0.0179·49-s − 0.0704·51-s − 1.32·53-s − 0.0859·55-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 \( 1 + 2.26T + 3T^{2} \)
5 \( 1 - 0.637T + 5T^{2} \)
7 \( 1 + 2.66T + 7T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 - 0.222T + 17T^{2} \)
23 \( 1 - 9.48T + 23T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 - 3.05T + 31T^{2} \)
37 \( 1 - 7.18T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 1.53T + 43T^{2} \)
47 \( 1 + 5.94T + 47T^{2} \)
53 \( 1 + 9.63T + 53T^{2} \)
59 \( 1 + 2.65T + 59T^{2} \)
61 \( 1 + 0.809T + 61T^{2} \)
67 \( 1 - 2.22T + 67T^{2} \)
71 \( 1 + 2.58T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 + 5.69T + 79T^{2} \)
83 \( 1 - 2.93T + 83T^{2} \)
89 \( 1 + 5.54T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−8.205386969548162684039517674971, −7.27109688375133514993966094677, −6.45722145089256775764225397865, −6.19029754109560955963051573101, −5.13597671929187469459073753526, −4.77753439191603489589619625981, −3.43440064349344977222718886072, −2.63982007731356149131265717136, −1.13242475080857161737610997008, 0, 1.13242475080857161737610997008, 2.63982007731356149131265717136, 3.43440064349344977222718886072, 4.77753439191603489589619625981, 5.13597671929187469459073753526, 6.19029754109560955963051573101, 6.45722145089256775764225397865, 7.27109688375133514993966094677, 8.205386969548162684039517674971

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