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Properties

Label 2-1536-12.11-c1-0-13
Degree $2$
Conductor $1536$
Sign $0.962 - 0.270i$
Analytic cond. $12.2650$
Root an. cond. $3.50214$
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  + (−0.468 − 1.66i)3-s − 3.02i·5-s + 3.62i·7-s + (−2.56 + 1.56i)9-s + 3.33·11-s − 1.32·13-s + (−5.03 + 1.41i)15-s + 7.12i·17-s + 5.20i·19-s + (6.04 − 1.69i)21-s + 4.41·23-s − 4.12·25-s + (3.80 + 3.54i)27-s + 6.41i·29-s − 0.794i·31-s + ⋯
L(s)  = 1  + (−0.270 − 0.962i)3-s − 1.35i·5-s + 1.36i·7-s + (−0.853 + 0.520i)9-s + 1.00·11-s − 0.367·13-s + (−1.30 + 0.365i)15-s + 1.72i·17-s + 1.19i·19-s + (1.31 − 0.370i)21-s + 0.920·23-s − 0.824·25-s + (0.731 + 0.681i)27-s + 1.19i·29-s − 0.142i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(12.2650\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.312099797\)
\(L(\frac12)\) \(\approx\) \(1.312099797\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.468 + 1.66i)T \)
good5 \( 1 + 3.02iT - 5T^{2} \)
7 \( 1 - 3.62iT - 7T^{2} \)
11 \( 1 - 3.33T + 11T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 - 7.12iT - 17T^{2} \)
19 \( 1 - 5.20iT - 19T^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 - 6.41iT - 29T^{2} \)
31 \( 1 + 0.794iT - 31T^{2} \)
37 \( 1 + 8.10T + 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + 3.76iT - 53T^{2} \)
59 \( 1 - 5.73T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 - 7.08iT - 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 4.86iT - 79T^{2} \)
83 \( 1 + 0.410T + 83T^{2} \)
89 \( 1 - 4.87iT - 89T^{2} \)
97 \( 1 - 1.12T + 97T^{2} \)
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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

−9.019540479678635802051302160900, −8.740754431479206117252277072255, −8.129189013333628390811141829585, −7.02644534536011867140808343390, −6.03769363618640804590562499391, −5.58232435092965943150030374680, −4.69503303452811014914520609945, −3.41608708835276663945070450315, −1.93978852424147912200137895146, −1.30382793602168933089095016035, 0.58134991202377865149537193575, 2.67580577977176634928150530971, 3.43555195173422361824347182171, 4.31647237278174974603064563825, 5.07986422576135130656966121828, 6.36899953771180856436691709129, 7.01784185841199525898999161532, 7.46244497366679321485024384567, 8.976927615104274652863163995035, 9.510739108048224811632859566923

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