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Properties

Label 2-1395-155.154-c0-0-1
Degree $2$
Conductor $1395$
Sign $-0.866 - 0.5i$
Analytic cond. $0.696195$
Root an. cond. $0.834383$
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 + (−0.866 − 0.5i)5-s + 1.73i·7-s + i·8-s + (0.5 − 0.866i)10-s − 1.73·14-s − 16-s − 19-s + (0.499 + 0.866i)25-s + 31-s + (0.866 − 1.49i)35-s − i·38-s + (0.5 − 0.866i)40-s − 1.73·41-s + 2i·47-s + ⋯
L(s)  = 1  + i·2-s + (−0.866 − 0.5i)5-s + 1.73i·7-s + i·8-s + (0.5 − 0.866i)10-s − 1.73·14-s − 16-s − 19-s + (0.499 + 0.866i)25-s + 31-s + (0.866 − 1.49i)35-s − i·38-s + (0.5 − 0.866i)40-s − 1.73·41-s + 2i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1395\)    =    \(3^{2} \cdot 5 \cdot 31\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(0.696195\)
Root analytic conductor: \(0.834383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1395} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ -0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9176251640\)
\(L(\frac12)\) \(\approx\) \(0.9176251640\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 - T \)
good2 \( 1 - iT - T^{2} \)
7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.73T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.864043641852318205944253179812, −8.725389721640801927535069173810, −8.542411789124516414095948300041, −7.78277838152655394686100273772, −6.76569236024476683979894803111, −6.05395188399216880919641661737, −5.25643968987634076840324152803, −4.51941576132058892709547804151, −3.07276822567712314524996848462, −2.02490715996413932437587672022, 0.74859907857198214897402552234, 2.19603340431485051861435453450, 3.47946381571401473622931261169, 3.87598988238509517333826105337, 4.79020396938036228741978635417, 6.60152144131854667197636183675, 6.87466042380464218859265212515, 7.77504275664237338150195385176, 8.590959696947837818931496981276, 10.01856107735145326953692356806

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