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Properties

Label 2-1395-155.114-c0-0-1
Degree $2$
Conductor $1395$
Sign $-0.950 + 0.311i$
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  + (1.14 − 1.58i)2-s + (−0.873 − 2.68i)4-s + (0.866 − 0.5i)5-s + (−3.39 − 1.10i)8-s + (0.204 − 1.94i)10-s + (−3.36 + 2.44i)16-s + (1.15 + 1.28i)17-s + (−0.190 + 0.0850i)19-s + (−2.10 − 1.89i)20-s + (−0.587 + 1.80i)23-s + (0.499 − 0.866i)25-s + (−0.809 − 0.587i)31-s + 4.57i·32-s + (3.36 − 0.354i)34-s + (−0.0850 + 0.400i)38-s + ⋯
L(s)  = 1  + (1.14 − 1.58i)2-s + (−0.873 − 2.68i)4-s + (0.866 − 0.5i)5-s + (−3.39 − 1.10i)8-s + (0.204 − 1.94i)10-s + (−3.36 + 2.44i)16-s + (1.15 + 1.28i)17-s + (−0.190 + 0.0850i)19-s + (−2.10 − 1.89i)20-s + (−0.587 + 1.80i)23-s + (0.499 − 0.866i)25-s + (−0.809 − 0.587i)31-s + 4.57i·32-s + (3.36 − 0.354i)34-s + (−0.0850 + 0.400i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1395\)    =    \(3^{2} \cdot 5 \cdot 31\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(0.696195\)
Root analytic conductor: \(0.834383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1395} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ -0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.007517375\)
\(L(\frac12)\) \(\approx\) \(2.007517375\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.14 + 1.58i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.913 - 0.406i)T^{2} \)
11 \( 1 + (0.104 + 0.994i)T^{2} \)
13 \( 1 + (0.669 + 0.743i)T^{2} \)
17 \( 1 + (-1.15 - 1.28i)T + (-0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.190 - 0.0850i)T + (0.669 - 0.743i)T^{2} \)
23 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.978 - 0.207i)T^{2} \)
43 \( 1 + (0.669 - 0.743i)T^{2} \)
47 \( 1 + (0.786 + 1.08i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (1.14 - 0.244i)T + (0.913 - 0.406i)T^{2} \)
59 \( 1 + (-0.978 + 0.207i)T^{2} \)
61 \( 1 + 0.813iT - T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.913 - 0.406i)T^{2} \)
73 \( 1 + (-0.104 - 0.994i)T^{2} \)
79 \( 1 + (-1.47 + 1.33i)T + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.155 - 1.47i)T + (-0.978 - 0.207i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.690758789616042988433121006450, −9.141274989665956400959280601945, −7.967060391426010734230412965837, −6.36831482528531204792223327743, −5.66971121861408318049852551320, −5.17032814128879767323915798951, −4.02446074946309044718686204725, −3.35332197253379620366978638324, −2.06496038853398806040436453495, −1.38321771434106497147179583054, 2.56558269092048975300097693210, 3.38218070445951819399517212907, 4.55581245992088985701064466121, 5.29265240809839484217211305715, 6.02544602190465031640159224697, 6.72599336002433910244790086762, 7.36951093580549400880379750459, 8.214668998062364213780010432955, 9.072831355541032485783245437963, 9.871123903892562113087527010276

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