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Properties

Label 2-1380-1380.1379-c0-0-7
Degree $2$
Conductor $1380$
Sign $-1$
Analytic cond. $0.688709$
Root an. cond. $0.829885$
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  − i·2-s − i·3-s − 4-s + 5-s − 6-s − 2i·7-s + i·8-s − 9-s − i·10-s + i·12-s − 2·14-s − i·15-s + 16-s + i·18-s − 20-s − 2·21-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s + 5-s − 6-s − 2i·7-s + i·8-s − 9-s − i·10-s + i·12-s − 2·14-s − i·15-s + 16-s + i·18-s − 20-s − 2·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078103164\)
\(L(\frac12)\) \(\approx\) \(1.078103164\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 - T \)
23 \( 1 - iT \)
good7 \( 1 + 2iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.581451351955579629399941807889, −8.691378844481420462814127647867, −7.68039968244768463823483492673, −7.08112760270616057233506832169, −6.06384825358104902791926739358, −5.10680297665112001570797653599, −4.01053279855729805336828427784, −3.03231755678777237983027733774, −1.80909134339008571648128408031, −0.966683351752560536549698484615, 2.26676310950139986127188412732, 3.28243228649763766269131907726, 4.75841923352050493681158572747, 5.22496741390739010348648279981, 6.05013742424902524587052455439, 6.46440997781863008629301781511, 8.059534057382555859767275245512, 8.730912250237001121336396595087, 9.237968133607540990562556198963, 9.802371500934351859117657799172

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