Properties

Label 2-3040-1.1-c1-0-64
Degree $2$
Conductor $3040$
Sign $-1$
Analytic cond. $24.2745$
Root an. cond. $4.92691$
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  + 2.68·3-s − 5-s − 3.20·7-s + 4.20·9-s − 6.44·11-s + 3.76·13-s − 2.68·15-s + 5.65·17-s + 19-s − 8.60·21-s − 7.20·23-s + 25-s + 3.24·27-s − 2.12·29-s − 10.8·31-s − 17.3·33-s + 3.20·35-s − 1.89·37-s + 10.1·39-s − 6.48·41-s + 5.49·43-s − 4.20·45-s + 2.86·47-s + 3.28·49-s + 15.1·51-s − 3.60·53-s + 6.44·55-s + ⋯
L(s)  = 1  + 1.54·3-s − 0.447·5-s − 1.21·7-s + 1.40·9-s − 1.94·11-s + 1.04·13-s − 0.693·15-s + 1.37·17-s + 0.229·19-s − 1.87·21-s − 1.50·23-s + 0.200·25-s + 0.623·27-s − 0.395·29-s − 1.95·31-s − 3.01·33-s + 0.542·35-s − 0.311·37-s + 1.61·39-s − 1.01·41-s + 0.837·43-s − 0.627·45-s + 0.417·47-s + 0.469·49-s + 2.12·51-s − 0.495·53-s + 0.869·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 - 2.68T + 3T^{2} \)
7 \( 1 + 3.20T + 7T^{2} \)
11 \( 1 + 6.44T + 11T^{2} \)
13 \( 1 - 3.76T + 13T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
23 \( 1 + 7.20T + 23T^{2} \)
29 \( 1 + 2.12T + 29T^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + 1.89T + 37T^{2} \)
41 \( 1 + 6.48T + 41T^{2} \)
43 \( 1 - 5.49T + 43T^{2} \)
47 \( 1 - 2.86T + 47T^{2} \)
53 \( 1 + 3.60T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 6.44T + 61T^{2} \)
67 \( 1 + 6.62T + 67T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 9.78T + 79T^{2} \)
83 \( 1 - 1.49T + 83T^{2} \)
89 \( 1 + 3.94T + 89T^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
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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

−8.101339486176022848792963817090, −7.87914284999458829008239175338, −7.19403803660374930689780925840, −6.02656234366650591509893354852, −5.33798853004363415490341309574, −3.95934216913525218572669779748, −3.38286339968352686063976977862, −2.87580278148223007771841052522, −1.79921411454893221575727194163, 0, 1.79921411454893221575727194163, 2.87580278148223007771841052522, 3.38286339968352686063976977862, 3.95934216913525218572669779748, 5.33798853004363415490341309574, 6.02656234366650591509893354852, 7.19403803660374930689780925840, 7.87914284999458829008239175338, 8.101339486176022848792963817090

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