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Properties

Label 2-1340-1340.439-c0-0-1
Degree $2$
Conductor $1340$
Sign $-0.103 - 0.994i$
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  + (0.5 + 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + 5-s + (0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (0.5 + 0.866i)10-s + (−0.499 + 0.866i)12-s − 0.999·14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)20-s + (−0.5 + 0.866i)21-s + (−0.5 − 0.866i)23-s − 0.999·24-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + 5-s + (0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (0.5 + 0.866i)10-s + (−0.499 + 0.866i)12-s − 0.999·14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)20-s + (−0.5 + 0.866i)21-s + (−0.5 − 0.866i)23-s − 0.999·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.103 - 0.994i$
Analytic conductor: \(0.668747\)
Root analytic conductor: \(0.817769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1340,\ (\ :0),\ -0.103 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.876002713\)
\(L(\frac12)\) \(\approx\) \(1.876002713\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
67 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 - T + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
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

−9.445285812689189705030608384677, −9.221102991251542765179990561638, −8.387912833464604364147073895771, −7.71289107201864159760155481873, −6.50360146453150046829595588620, −6.01918897584731669672191629193, −5.18055286267784535780108859804, −4.03434123891012771635414851328, −2.87851227751421674504024548412, −2.33794813529638694579695186463, 1.42090542391337303710290631651, 2.53230674281552409223651822622, 3.29462868440346362589522284430, 4.16153076640262034221703955182, 5.30578119359911596009224134553, 6.13560666009879403090114854446, 7.08494920302017735077720678127, 8.219967837409209907582219131549, 9.142856063372296779826422551292, 9.605619849608076550668132184410

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