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Properties

Label 2-1395-155.34-c0-0-0
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: $\chi_{1395} (964, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ 0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3587916347\)
\(L(\frac12)\) \(\approx\) \(0.3587916347\)
\(L(1)\) 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

−9.718528463215976698887362048270, −8.866312651383083842484417947035, −8.583981841155186152696134142088, −7.64593924313632933945410206539, −6.98564480455539845463625120600, −5.21488955110812755039452103635, −4.00887375880219649679988974538, −3.64594087260880848747865435757, −2.31819265351726572173298290330, −1.22654746925401529458044297993, 0.49893823318908562973077203283, 2.43793022805784947019600625484, 4.25848632725550748990120771665, 4.94656873763951909762863286545, 6.10393849324625650327500016606, 6.82833562997214463160980438180, 7.33069073108478319826264395433, 8.101059533480309683909827829335, 8.884811837168166706673845453868, 9.365092947526014744959590570419

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