Properties

Degree $2$
Conductor $2013$
Sign $-0.794 + 0.606i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 0.951i)2-s + (0.309 − 0.951i)3-s + (0.500 + 1.53i)4-s + (−1.30 + 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + 1.61·12-s + (1.30 + 0.951i)13-s + (0.5 − 0.363i)17-s + (0.499 + 1.53i)18-s + (0.190 − 0.587i)19-s + (−0.499 + 1.53i)22-s + 1.61·23-s + (−0.809 − 0.587i)24-s + (0.309 − 0.951i)25-s + ⋯
L(s)  = 1  + (−1.30 − 0.951i)2-s + (0.309 − 0.951i)3-s + (0.500 + 1.53i)4-s + (−1.30 + 0.951i)6-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + 1.61·12-s + (1.30 + 0.951i)13-s + (0.5 − 0.363i)17-s + (0.499 + 1.53i)18-s + (0.190 − 0.587i)19-s + (−0.499 + 1.53i)22-s + 1.61·23-s + (−0.809 − 0.587i)24-s + (0.309 − 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.794 + 0.606i$
Motivic weight: \(0\)
Character: $\chi_{2013} (1280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2013,\ (\ :0),\ -0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6786687788\)
\(L(\frac12)\) \(\approx\) \(0.6786687788\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + 0.618T + T^{2} \)
97 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.903194807418376849525201522610, −8.550252235458159037404463990648, −7.79404281807469457728038352139, −6.90504183356206748525713540895, −6.22742081900529262251414177567, −5.00506781233248800967592266795, −3.32256207049814413675247257431, −2.95776512549937567139967015216, −1.67533993624270204581236008400, −0.867892141232920152992376533936, 1.31947140726701015910000221118, 2.93517158591796632213683886653, 3.90780695540615163050013160301, 5.11476482861973461536635262930, 5.77442805733415641430982756458, 6.65271508152329928500705887755, 7.70378066040070171338932039772, 8.097399499973408895015522850456, 8.840752140108718286890006047932, 9.547504508010884349165009006049

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