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Properties

Label 2-1395-155.84-c0-0-1
Degree $2$
Conductor $1395$
Sign $-0.255 + 0.966i$
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  + (−1.73 − 0.564i)2-s + (1.89 + 1.37i)4-s + (0.866 − 0.5i)5-s + (−1.43 − 1.97i)8-s + (−1.78 + 0.379i)10-s + (0.657 + 2.02i)16-s + (−0.181 − 1.72i)17-s + (−1.30 − 1.45i)19-s + (2.32 + 0.244i)20-s + (−0.951 + 0.690i)23-s + (0.499 − 0.866i)25-s + (0.309 − 0.951i)31-s − 1.44i·32-s + (−0.657 + 3.09i)34-s + (1.45 + 3.26i)38-s + ⋯
L(s)  = 1  + (−1.73 − 0.564i)2-s + (1.89 + 1.37i)4-s + (0.866 − 0.5i)5-s + (−1.43 − 1.97i)8-s + (−1.78 + 0.379i)10-s + (0.657 + 2.02i)16-s + (−0.181 − 1.72i)17-s + (−1.30 − 1.45i)19-s + (2.32 + 0.244i)20-s + (−0.951 + 0.690i)23-s + (0.499 − 0.866i)25-s + (0.309 − 0.951i)31-s − 1.44i·32-s + (−0.657 + 3.09i)34-s + (1.45 + 3.26i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1395\)    =    \(3^{2} \cdot 5 \cdot 31\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(0.696195\)
Root analytic conductor: \(0.834383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1395} (1324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ -0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4999931871\)
\(L(\frac12)\) \(\approx\) \(0.4999931871\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (1.73 + 0.564i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.669 + 0.743i)T^{2} \)
11 \( 1 + (0.978 + 0.207i)T^{2} \)
13 \( 1 + (-0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.181 + 1.72i)T + (-0.978 + 0.207i)T^{2} \)
19 \( 1 + (1.30 + 1.45i)T + (-0.104 + 0.994i)T^{2} \)
23 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.913 - 0.406i)T^{2} \)
43 \( 1 + (-0.104 + 0.994i)T^{2} \)
47 \( 1 + (-0.198 + 0.0646i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-1.73 - 0.773i)T + (0.669 + 0.743i)T^{2} \)
59 \( 1 + (0.913 + 0.406i)T^{2} \)
61 \( 1 - 1.48iT - T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.669 + 0.743i)T^{2} \)
73 \( 1 + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (0.413 - 0.0434i)T + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (-1.94 + 0.413i)T + (0.913 - 0.406i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.352431537271967239532207057919, −9.046337249605102941886893975351, −8.242016286661895350301073032470, −7.29049105358396571653905080438, −6.63207114651940833579893684857, −5.52880953479706391617679784533, −4.31590743328454196532454092920, −2.69550853914072783724452242755, −2.13314114167966396063734874615, −0.72036563384724539142926072170, 1.59806168371696723471289506951, 2.29513139552747459777037206851, 3.91502195282179093597396140604, 5.58691632952527684318113629320, 6.31117726187482196890489603046, 6.67009590782123227718196650432, 7.88139458370241671693032186019, 8.384958468532026152331626051954, 9.092274207682087468226434394536, 10.07715853817797943318665371458

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