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Properties

Label 2-1536-3.2-c0-0-4
Degree $2$
Conductor $1536$
Sign $1$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
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  + i·3-s − 1.41i·5-s + 1.41·7-s − 9-s + 1.41·15-s + 1.41i·21-s − 1.00·25-s − i·27-s − 1.41i·29-s + 1.41·31-s − 2.00i·35-s + 1.41i·45-s + 1.00·49-s + 1.41i·53-s + 2i·59-s + ⋯
L(s)  = 1  + i·3-s − 1.41i·5-s + 1.41·7-s − 9-s + 1.41·15-s + 1.41i·21-s − 1.00·25-s − i·27-s − 1.41i·29-s + 1.41·31-s − 2.00i·35-s + 1.41i·45-s + 1.00·49-s + 1.41i·53-s + 2i·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $1$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.245878109\)
\(L(\frac12)\) \(\approx\) \(1.245878109\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.592918515533776832436371648227, −8.733527355938268763248809825528, −8.351091465986634389911256902067, −7.59389299902587054844093715112, −6.02447825685202347727637113729, −5.30535955489511374760814749431, −4.52003089030410023500199749084, −4.19659847462567164862127246443, −2.59566197107718367984931429705, −1.21128091372142984625068142395, 1.50900546726229805037942865789, 2.47918794121677231774986272014, 3.38582658671152759664466590645, 4.76938348563243513634370399072, 5.68953661969341963097242831986, 6.70469351939790803109367351248, 7.10221005944014512368860071881, 8.059751988466862252190572118035, 8.448265334211551771438250045028, 9.713942461936846378110088527104

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