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Properties

Label 2-1340-1340.559-c0-0-1
Degree $2$
Conductor $1340$
Sign $0.909 + 0.414i$
Analytic cond. $0.668747$
Root an. cond. $0.817769$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.235 + 0.971i)2-s + (0.252 − 1.75i)3-s + (−0.888 + 0.458i)4-s + (0.415 + 0.909i)5-s + (1.76 − 0.168i)6-s + (1.34 − 1.28i)7-s + (−0.654 − 0.755i)8-s + (−2.07 − 0.608i)9-s + (−0.786 + 0.618i)10-s + (0.581 + 1.67i)12-s + (1.56 + 1.00i)14-s + (1.70 − 0.500i)15-s + (0.580 − 0.814i)16-s + (0.102 − 2.15i)18-s + (−0.786 − 0.618i)20-s + (−1.91 − 2.68i)21-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)2-s + (0.252 − 1.75i)3-s + (−0.888 + 0.458i)4-s + (0.415 + 0.909i)5-s + (1.76 − 0.168i)6-s + (1.34 − 1.28i)7-s + (−0.654 − 0.755i)8-s + (−2.07 − 0.608i)9-s + (−0.786 + 0.618i)10-s + (0.581 + 1.67i)12-s + (1.56 + 1.00i)14-s + (1.70 − 0.500i)15-s + (0.580 − 0.814i)16-s + (0.102 − 2.15i)18-s + (−0.786 − 0.618i)20-s + (−1.91 − 2.68i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.909 + 0.414i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ 0.909 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368308201\)
\(L(\frac12)\) \(\approx\) \(1.368308201\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.235 - 0.971i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.0475 + 0.998i)T \)
good3 \( 1 + (-0.252 + 1.75i)T + (-0.959 - 0.281i)T^{2} \)
7 \( 1 + (-1.34 + 1.28i)T + (0.0475 - 0.998i)T^{2} \)
11 \( 1 + (-0.981 - 0.189i)T^{2} \)
13 \( 1 + (-0.928 - 0.371i)T^{2} \)
17 \( 1 + (-0.580 - 0.814i)T^{2} \)
19 \( 1 + (-0.0475 - 0.998i)T^{2} \)
23 \( 1 + (0.928 - 0.371i)T + (0.723 - 0.690i)T^{2} \)
29 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.928 + 0.371i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.0934 - 1.96i)T + (-0.995 + 0.0950i)T^{2} \)
43 \( 1 + (-0.0800 + 0.0514i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.514 - 0.404i)T + (0.235 + 0.971i)T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.827 - 0.0789i)T + (0.981 - 0.189i)T^{2} \)
71 \( 1 + (-0.580 + 0.814i)T^{2} \)
73 \( 1 + (-0.981 + 0.189i)T^{2} \)
79 \( 1 + (0.786 - 0.618i)T^{2} \)
83 \( 1 + (-0.839 + 1.17i)T + (-0.327 - 0.945i)T^{2} \)
89 \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.583438866107151967723490069430, −8.346442702890765528002610979590, −7.69501306251626218306063958351, −7.53602969862797236481437602350, −6.50341742959205792804803288576, −6.12116618226899675048541463740, −4.87366314240502644974893482311, −3.72943825850722333907750361549, −2.43425111801727657839615423428, −1.20922598231804179553607016683, 1.83065906616659269374068496262, 2.76798226930760143304906187787, 4.02351527897609514428383924378, 4.69131924694994237710092815603, 5.32370401967610376577225058026, 5.78714367130296226772044865402, 8.137497366046613427754257289244, 8.815967477388806935035023077207, 8.978320633841033191012097818186, 9.938452803168832548023412984277

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