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Properties

Label 2-1007-1.1-c1-0-75
Degree $2$
Conductor $1007$
Sign $-1$
Analytic cond. $8.04093$
Root an. cond. $2.83565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 3·5-s − 7-s − 3·9-s − 6·10-s + 3·11-s − 2·14-s − 4·16-s − 3·17-s − 6·18-s − 19-s − 6·20-s + 6·22-s − 8·23-s + 4·25-s − 2·28-s + 2·29-s − 2·31-s − 8·32-s − 6·34-s + 3·35-s − 6·36-s − 4·37-s − 2·38-s + 6·41-s − 5·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.34·5-s − 0.377·7-s − 9-s − 1.89·10-s + 0.904·11-s − 0.534·14-s − 16-s − 0.727·17-s − 1.41·18-s − 0.229·19-s − 1.34·20-s + 1.27·22-s − 1.66·23-s + 4/5·25-s − 0.377·28-s + 0.371·29-s − 0.359·31-s − 1.41·32-s − 1.02·34-s + 0.507·35-s − 36-s − 0.657·37-s − 0.324·38-s + 0.937·41-s − 0.762·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $-1$
Analytic conductor: \(8.04093\)
Root analytic conductor: \(2.83565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1007,\ (\ :1/2),\ -1)\)

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)$
bad19 \( 1 + T \)
53 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.433548470637644072669686071348, −8.626470330066807396264615662716, −7.78997306487226182646384131671, −6.65430570134456255086455486851, −6.09174406445201527513865362791, −5.00007502799761395288419220728, −3.98790135972688185122818775592, −3.62666575170944973731750940171, −2.43502883288959516917088648342, 0, 2.43502883288959516917088648342, 3.62666575170944973731750940171, 3.98790135972688185122818775592, 5.00007502799761395288419220728, 6.09174406445201527513865362791, 6.65430570134456255086455486851, 7.78997306487226182646384131671, 8.626470330066807396264615662716, 9.433548470637644072669686071348

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