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Properties

Label 2-1452-11.3-c1-0-17
Degree $2$
Conductor $1452$
Sign $-0.834 - 0.551i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (−1.61 − 1.17i)5-s + (−0.618 + 1.90i)7-s + (−0.809 + 0.587i)9-s + (4.85 − 3.52i)13-s + (−0.618 + 1.90i)15-s + (−3.23 − 2.35i)17-s + (0.618 + 1.90i)19-s + 1.99·21-s − 8·23-s + (−0.309 − 0.951i)25-s + (0.809 + 0.587i)27-s + (3.23 − 2.35i)35-s + (−1.85 + 5.70i)37-s + (−4.85 − 3.52i)39-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (−0.723 − 0.525i)5-s + (−0.233 + 0.718i)7-s + (−0.269 + 0.195i)9-s + (1.34 − 0.978i)13-s + (−0.159 + 0.491i)15-s + (−0.784 − 0.570i)17-s + (0.141 + 0.436i)19-s + 0.436·21-s − 1.66·23-s + (−0.0618 − 0.190i)25-s + (0.155 + 0.113i)27-s + (0.546 − 0.397i)35-s + (−0.304 + 0.938i)37-s + (−0.777 − 0.564i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.834 - 0.551i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ -0.834 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06662637079\)
\(L(\frac12)\) \(\approx\) \(0.06662637079\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (1.61 + 1.17i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.618 - 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.85 + 3.52i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.23 + 2.35i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.618 - 1.90i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.85 - 5.70i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.3 - 8.22i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.70 - 11.4i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (11.3 + 8.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.85 - 5.70i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.61 + 1.17i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.9 - 9.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-1.61 + 1.17i)T + (29.9 - 92.2i)T^{2} \)
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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

−8.771490015997783520558472726206, −8.244273452418885433429665835278, −7.65265625299180822568844604231, −6.39613506957927144785057626322, −5.91936527365518944800675490060, −4.88582864493712095474751615180, −3.84869435931799853250734361138, −2.82687517947176178896265976086, −1.47692177525037913695599699821, −0.02742191199574877987703514672, 1.83241444872562866534214653331, 3.50628361182230876993752462292, 3.86317278757586093838790983888, 4.75956753699569729149043202311, 6.12812579665279411027329284776, 6.62866924948525863915951286455, 7.58529699192288290618175732433, 8.427404083066443195442390907462, 9.207313571232388860358871031646, 10.10600506873494688183796243275

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