Properties

Label 2-3344-1.1-c1-0-33
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  + 3.22·3-s − 1.96·5-s − 3.83·7-s + 7.39·9-s − 11-s − 4.08·13-s − 6.31·15-s + 6.84·17-s + 19-s − 12.3·21-s + 6.67·23-s − 1.15·25-s + 14.1·27-s + 3.43·29-s + 5.74·31-s − 3.22·33-s + 7.52·35-s + 4.55·37-s − 13.1·39-s − 0.970·41-s + 6.42·43-s − 14.4·45-s − 7.92·47-s + 7.71·49-s + 22.0·51-s + 13.4·53-s + 1.96·55-s + ⋯
L(s)  = 1  + 1.86·3-s − 0.876·5-s − 1.45·7-s + 2.46·9-s − 0.301·11-s − 1.13·13-s − 1.63·15-s + 1.65·17-s + 0.229·19-s − 2.69·21-s + 1.39·23-s − 0.231·25-s + 2.72·27-s + 0.638·29-s + 1.03·31-s − 0.561·33-s + 1.27·35-s + 0.748·37-s − 2.10·39-s − 0.151·41-s + 0.980·43-s − 2.15·45-s − 1.15·47-s + 1.10·49-s + 3.08·51-s + 1.84·53-s + 0.264·55-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

\(L(1)\) \(\approx\) \(2.764010120\)
\(L(\frac12)\) \(\approx\) \(2.764010120\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.22T + 3T^{2} \)
5 \( 1 + 1.96T + 5T^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
13 \( 1 + 4.08T + 13T^{2} \)
17 \( 1 - 6.84T + 17T^{2} \)
23 \( 1 - 6.67T + 23T^{2} \)
29 \( 1 - 3.43T + 29T^{2} \)
31 \( 1 - 5.74T + 31T^{2} \)
37 \( 1 - 4.55T + 37T^{2} \)
41 \( 1 + 0.970T + 41T^{2} \)
43 \( 1 - 6.42T + 43T^{2} \)
47 \( 1 + 7.92T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 9.76T + 61T^{2} \)
67 \( 1 + 9.19T + 67T^{2} \)
71 \( 1 - 7.97T + 71T^{2} \)
73 \( 1 - 4.01T + 73T^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 7.57T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−8.476898457629414529658872896545, −7.995539059804957349244168304358, −7.20614456425244259455709584775, −6.90868705414665306603477020899, −5.52644099199773919816633736591, −4.42103309913710091548749566935, −3.65501873864957942141160721198, −2.98016536488151245915654880890, −2.57356105840017217536798557930, −0.910548424048769001036091130302, 0.910548424048769001036091130302, 2.57356105840017217536798557930, 2.98016536488151245915654880890, 3.65501873864957942141160721198, 4.42103309913710091548749566935, 5.52644099199773919816633736591, 6.90868705414665306603477020899, 7.20614456425244259455709584775, 7.995539059804957349244168304358, 8.476898457629414529658872896545

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