↑

Properties

Label 2-1536-24.5-c0-0-6
Degree $2$
Conductor $1536$
Sign $0.707 + 0.707i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)3-s − 1.00i·9-s + 1.41·11-s + 2i·17-s − 1.41i·19-s − 25-s + (−0.707 − 0.707i)27-s + (1.00 − 1.00i)33-s − 1.41i·43-s − 49-s + (1.41 + 1.41i)51-s + (−1.00 − 1.00i)57-s + 1.41·59-s + 1.41i·67-s + (−0.707 + 0.707i)75-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s − 1.00i·9-s + 1.41·11-s + 2i·17-s − 1.41i·19-s − 25-s + (−0.707 − 0.707i)27-s + (1.00 − 1.00i)33-s − 1.41i·43-s − 49-s + (1.41 + 1.41i)51-s + (−1.00 − 1.00i)57-s + 1.41·59-s + 1.41i·67-s + (−0.707 + 0.707i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.468849489\)
\(L(\frac12)\) \(\approx\) \(1.468849489\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.341401595658325147369657805807, −8.686469330731870426588050634831, −8.096813780344601415523330170015, −7.03047285266111657885124206297, −6.53113102376732973402996882877, −5.66300299032654620468513801698, −4.16725213998745009468642844605, −3.61922135409360049796313843150, −2.31177929859671629348564720464, −1.33799301563258911267453397070, 1.65769187283476453183726702078, 2.92341292824868345103507992255, 3.79315761007029511398875558093, 4.56466751642100304596034654761, 5.53624487374349425541996904315, 6.55695060222997539726873604225, 7.51331925178072525887364451906, 8.211345046660685570147794725861, 9.178925736611971243311118863559, 9.596901210332445979528196728386

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