Properties

Label 2-3344-1.1-c1-0-72
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  + 2.93·3-s + 3.77·5-s + 2.25·7-s + 5.59·9-s − 11-s − 1.48·13-s + 11.0·15-s + 3.18·17-s + 19-s + 6.61·21-s − 8.68·23-s + 9.26·25-s + 7.60·27-s − 9.28·29-s + 3.52·31-s − 2.93·33-s + 8.51·35-s + 8.84·37-s − 4.34·39-s − 10.9·41-s − 5.47·43-s + 21.1·45-s + 3.55·47-s − 1.91·49-s + 9.35·51-s − 0.536·53-s − 3.77·55-s + ⋯
L(s)  = 1  + 1.69·3-s + 1.68·5-s + 0.852·7-s + 1.86·9-s − 0.301·11-s − 0.411·13-s + 2.85·15-s + 0.773·17-s + 0.229·19-s + 1.44·21-s − 1.81·23-s + 1.85·25-s + 1.46·27-s − 1.72·29-s + 0.633·31-s − 0.510·33-s + 1.43·35-s + 1.45·37-s − 0.695·39-s − 1.70·41-s − 0.834·43-s + 3.14·45-s + 0.518·47-s − 0.273·49-s + 1.30·51-s − 0.0736·53-s − 0.509·55-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

\(L(1)\) \(\approx\) \(5.175029848\)
\(L(\frac12)\) \(\approx\) \(5.175029848\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.93T + 3T^{2} \)
5 \( 1 - 3.77T + 5T^{2} \)
7 \( 1 - 2.25T + 7T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 - 3.18T + 17T^{2} \)
23 \( 1 + 8.68T + 23T^{2} \)
29 \( 1 + 9.28T + 29T^{2} \)
31 \( 1 - 3.52T + 31T^{2} \)
37 \( 1 - 8.84T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 5.47T + 43T^{2} \)
47 \( 1 - 3.55T + 47T^{2} \)
53 \( 1 + 0.536T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 8.03T + 61T^{2} \)
67 \( 1 - 8.22T + 67T^{2} \)
71 \( 1 + 6.42T + 71T^{2} \)
73 \( 1 - 1.66T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 5.79T + 83T^{2} \)
89 \( 1 - 3.88T + 89T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−8.622327681096374000919671255695, −7.945571525261231418028970224958, −7.47688755413831939090196755028, −6.36368814439777764585181785488, −5.56774280073741397450985089912, −4.78816237440784302655237820112, −3.74887031567320762677097148995, −2.80090459947056070764888378042, −2.02157741303540660339650693164, −1.56846475104487573032851499350, 1.56846475104487573032851499350, 2.02157741303540660339650693164, 2.80090459947056070764888378042, 3.74887031567320762677097148995, 4.78816237440784302655237820112, 5.56774280073741397450985089912, 6.36368814439777764585181785488, 7.47688755413831939090196755028, 7.945571525261231418028970224958, 8.622327681096374000919671255695

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