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Properties

Label 2-1395-155.44-c0-0-1
Degree $2$
Conductor $1395$
Sign $0.561 - 0.827i$
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  + (0.786 + 1.08i)2-s + (−0.244 + 0.752i)4-s + (0.866 − 0.5i)5-s + (0.266 − 0.0864i)8-s + (1.22 + 0.544i)10-s + (0.942 + 0.684i)16-s + (−1.69 − 0.360i)17-s + (−0.190 − 1.81i)19-s + (0.164 + 0.773i)20-s + (0.587 + 1.80i)23-s + (0.499 − 0.866i)25-s + (−0.809 + 0.587i)31-s + 1.27i·32-s + (−0.942 − 2.11i)34-s + (1.81 − 1.63i)38-s + ⋯
L(s)  = 1  + (0.786 + 1.08i)2-s + (−0.244 + 0.752i)4-s + (0.866 − 0.5i)5-s + (0.266 − 0.0864i)8-s + (1.22 + 0.544i)10-s + (0.942 + 0.684i)16-s + (−1.69 − 0.360i)17-s + (−0.190 − 1.81i)19-s + (0.164 + 0.773i)20-s + (0.587 + 1.80i)23-s + (0.499 − 0.866i)25-s + (−0.809 + 0.587i)31-s + 1.27i·32-s + (−0.942 − 2.11i)34-s + (1.81 − 1.63i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1395\)    =    \(3^{2} \cdot 5 \cdot 31\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(0.696195\)
Root analytic conductor: \(0.834383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1395} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.887233895\)
\(L(\frac12)\) \(\approx\) \(1.887233895\)
\(L(1)\) 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 \( 1 + (-0.786 - 1.08i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.104 - 0.994i)T^{2} \)
11 \( 1 + (-0.913 + 0.406i)T^{2} \)
13 \( 1 + (-0.978 - 0.207i)T^{2} \)
17 \( 1 + (1.69 + 0.360i)T + (0.913 + 0.406i)T^{2} \)
19 \( 1 + (0.190 + 1.81i)T + (-0.978 + 0.207i)T^{2} \)
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 \( 1 + (0.669 + 0.743i)T^{2} \)
43 \( 1 + (-0.978 + 0.207i)T^{2} \)
47 \( 1 + (1.14 - 1.58i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.786 - 0.873i)T + (-0.104 - 0.994i)T^{2} \)
59 \( 1 + (0.669 - 0.743i)T^{2} \)
61 \( 1 + 1.98iT - T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.104 - 0.994i)T^{2} \)
73 \( 1 + (0.913 - 0.406i)T^{2} \)
79 \( 1 + (0.169 - 0.795i)T + (-0.913 - 0.406i)T^{2} \)
83 \( 1 + (0.379 + 0.169i)T + (0.669 + 0.743i)T^{2} \)
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

−9.444129575291171170962552933839, −9.203450294021817115437763253735, −8.119462482182695002139189787259, −7.08683291984731881084134406262, −6.61220678306239248119962451078, −5.72330163938020639784681532411, −4.92695312120542654558915179053, −4.45013776782865559027582872445, −2.98560926016406831124741422335, −1.62629419802424066883045179627, 1.78685079364853488984205446620, 2.39878145483786845480513690121, 3.50117121470775822311743515296, 4.34495410008099517319005825924, 5.28595686488346568925801455495, 6.22387745922045333380471278923, 6.95253936667295464822263682427, 8.169700150766867208332195508767, 8.991808045344404966314177678012, 10.13623030816968785574181915326

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