Properties

Label 2-1340-1340.559-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5598928833\)
\(L(\frac12)\) \(\approx\) \(0.5598928833\)
\(L(1)\) 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

−9.934765889606996147082794528067, −9.711200885746949095780310898145, −9.076483511052360847656217553165, −8.247531455981116314632285463252, −6.58088005759572067000099683920, −5.81637488208804418258149215283, −4.97222133785271703997658571925, −3.91846343789192635513923500078, −2.99041832788726288776613922191, −2.63218118486914204061693956864, 0.54464371864228381590898227167, 1.50942914130879478707393859834, 3.29618662307235288693225438206, 4.67586452816510036530151544873, 5.70644922790998691374237650326, 6.37239016604750994704531693482, 7.09620480835076725396801412063, 7.46290177658834061441581439548, 8.524287067344313638895665261081, 9.142948433497807690315934317120

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