Properties

Label 2-3344-1.1-c1-0-62
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  + 1.87·3-s + 2.83·5-s + 3.83·7-s + 0.498·9-s + 11-s + 6.12·13-s + 5.29·15-s − 3.70·17-s − 19-s + 7.16·21-s − 1.01·23-s + 3.02·25-s − 4.67·27-s − 3.56·29-s + 5.29·31-s + 1.87·33-s + 10.8·35-s + 1.68·37-s + 11.4·39-s − 6.71·41-s + 3.59·43-s + 1.41·45-s + 6.48·47-s + 7.69·49-s − 6.93·51-s − 6.34·53-s + 2.83·55-s + ⋯
L(s)  = 1  + 1.07·3-s + 1.26·5-s + 1.44·7-s + 0.166·9-s + 0.301·11-s + 1.69·13-s + 1.36·15-s − 0.899·17-s − 0.229·19-s + 1.56·21-s − 0.212·23-s + 0.605·25-s − 0.900·27-s − 0.662·29-s + 0.951·31-s + 0.325·33-s + 1.83·35-s + 0.277·37-s + 1.83·39-s − 1.04·41-s + 0.548·43-s + 0.210·45-s + 0.945·47-s + 1.09·49-s − 0.971·51-s − 0.872·53-s + 0.382·55-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

\(L(1)\) \(\approx\) \(4.334723356\)
\(L(\frac12)\) \(\approx\) \(4.334723356\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.87T + 3T^{2} \)
5 \( 1 - 2.83T + 5T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
13 \( 1 - 6.12T + 13T^{2} \)
17 \( 1 + 3.70T + 17T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 - 5.29T + 31T^{2} \)
37 \( 1 - 1.68T + 37T^{2} \)
41 \( 1 + 6.71T + 41T^{2} \)
43 \( 1 - 3.59T + 43T^{2} \)
47 \( 1 - 6.48T + 47T^{2} \)
53 \( 1 + 6.34T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 0.743T + 61T^{2} \)
67 \( 1 + 2.07T + 67T^{2} \)
71 \( 1 - 2.65T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 - 1.81T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 1.51T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−8.566586289116982170861515972938, −8.226674235382473632492461053914, −7.27318766235173818187019682962, −6.19581209991098071507790604596, −5.76692882760851629555477482290, −4.68909437009930878556763349813, −3.92719954471507299255435539368, −2.85711174793307199484108071881, −1.92957895989215285813473565454, −1.43190378759237055537892297456, 1.43190378759237055537892297456, 1.92957895989215285813473565454, 2.85711174793307199484108071881, 3.92719954471507299255435539368, 4.68909437009930878556763349813, 5.76692882760851629555477482290, 6.19581209991098071507790604596, 7.27318766235173818187019682962, 8.226674235382473632492461053914, 8.566586289116982170861515972938

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