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Properties

Label 2-1536-12.11-c1-0-16
Degree $2$
Conductor $1536$
Sign $-i$
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  + 1.73·3-s − 0.449i·5-s + 4.87i·7-s + 2.99·9-s − 5.65·11-s − 0.778i·15-s + 8.44i·21-s + 4.79·25-s + 5.19·27-s + 9.34i·29-s + 10.5i·31-s − 9.79·33-s + 2.19·35-s − 1.34i·45-s − 16.7·49-s + ⋯
L(s)  = 1  + 1.00·3-s − 0.201i·5-s + 1.84i·7-s + 0.999·9-s − 1.70·11-s − 0.201i·15-s + 1.84i·21-s + 0.959·25-s + 1.00·27-s + 1.73i·29-s + 1.89i·31-s − 1.70·33-s + 0.370·35-s − 0.201i·45-s − 2.39·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-i$
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: \((2,\ 1536,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.987163307\)
\(L(\frac12)\) \(\approx\) \(1.987163307\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
good5 \( 1 + 0.449iT - 5T^{2} \)
7 \( 1 - 4.87iT - 7T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.34iT - 29T^{2} \)
31 \( 1 - 10.5iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 3.32iT - 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.408385322113007482374827692533, −8.682640430356627394086793757763, −8.387901944313939300496653574509, −7.43412606055423976850303958061, −6.47521314279419432174891654539, −5.18776584295219361818025992114, −5.00140388549961118794277183796, −3.24127045718402932051214819100, −2.74027908252705492380406356654, −1.75128086306209944300041680019, 0.66367754802862313734761721297, 2.22212210539114712435664602942, 3.15331094072632467554271899914, 4.12572545227499583080858695731, 4.76395012377382587670637965927, 6.14055539697587770335382785561, 7.19539953406741669101035383896, 7.73344742135372097297616663823, 8.133018879997777469858128574870, 9.393654776705339887830251878068

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