Properties

Label 2-3344-1.1-c1-0-67
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  − 0.121·3-s − 1.06·5-s + 3.36·7-s − 2.98·9-s − 11-s − 2.15·13-s + 0.129·15-s + 3.67·17-s + 19-s − 0.410·21-s − 3.15·23-s − 3.87·25-s + 0.729·27-s + 7.17·29-s − 4.65·31-s + 0.121·33-s − 3.57·35-s + 2.27·37-s + 0.263·39-s − 11.3·41-s − 9.38·43-s + 3.16·45-s + 5.77·47-s + 4.34·49-s − 0.447·51-s − 5.65·53-s + 1.06·55-s + ⋯
L(s)  = 1  − 0.0703·3-s − 0.474·5-s + 1.27·7-s − 0.995·9-s − 0.301·11-s − 0.599·13-s + 0.0333·15-s + 0.890·17-s + 0.229·19-s − 0.0895·21-s − 0.657·23-s − 0.775·25-s + 0.140·27-s + 1.33·29-s − 0.835·31-s + 0.0212·33-s − 0.603·35-s + 0.373·37-s + 0.0421·39-s − 1.77·41-s − 1.43·43-s + 0.471·45-s + 0.841·47-s + 0.621·49-s − 0.0626·51-s − 0.777·53-s + 0.142·55-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 \( 1 + 0.121T + 3T^{2} \)
5 \( 1 + 1.06T + 5T^{2} \)
7 \( 1 - 3.36T + 7T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
17 \( 1 - 3.67T + 17T^{2} \)
23 \( 1 + 3.15T + 23T^{2} \)
29 \( 1 - 7.17T + 29T^{2} \)
31 \( 1 + 4.65T + 31T^{2} \)
37 \( 1 - 2.27T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 9.38T + 43T^{2} \)
47 \( 1 - 5.77T + 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 6.98T + 61T^{2} \)
67 \( 1 - 4.81T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 8.08T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 9.96T + 83T^{2} \)
89 \( 1 + 4.61T + 89T^{2} \)
97 \( 1 + 4.09T + 97T^{2} \)
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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

−8.292321267299929268411380810900, −7.67007312318362361394335551467, −6.90482643658369247486092256144, −5.74635692541419623604999287060, −5.25158298423880276646228679523, −4.47609912087358215839333661669, −3.48937621509181508777761457380, −2.55087113484843366386239488627, −1.48069889285261637294403344002, 0, 1.48069889285261637294403344002, 2.55087113484843366386239488627, 3.48937621509181508777761457380, 4.47609912087358215839333661669, 5.25158298423880276646228679523, 5.74635692541419623604999287060, 6.90482643658369247486092256144, 7.67007312318362361394335551467, 8.292321267299929268411380810900

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