Properties

Label 2-1536-8.5-c1-0-4
Degree $2$
Conductor $1536$
Sign $-1$
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  + i·3-s + 3.41i·5-s + 0.585·7-s − 9-s − 2i·11-s + 2.82i·13-s − 3.41·15-s − 7.65·17-s + 5.65i·19-s + 0.585i·21-s + 6.82·23-s − 6.65·25-s − i·27-s − 3.41i·29-s − 7.41·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.52i·5-s + 0.221·7-s − 0.333·9-s − 0.603i·11-s + 0.784i·13-s − 0.881·15-s − 1.85·17-s + 1.29i·19-s + 0.127i·21-s + 1.42·23-s − 1.33·25-s − 0.192i·27-s − 0.634i·29-s − 1.33·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-1$
Analytic conductor: \(12.2650\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.044451636\)
\(L(\frac12)\) \(\approx\) \(1.044451636\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 - 3.41iT - 5T^{2} \)
7 \( 1 - 0.585T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 - 5.65iT - 19T^{2} \)
23 \( 1 - 6.82T + 23T^{2} \)
29 \( 1 + 3.41iT - 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 - 1.65iT - 37T^{2} \)
41 \( 1 - 0.343T + 41T^{2} \)
43 \( 1 - 9.65iT - 43T^{2} \)
47 \( 1 - 4.48T + 47T^{2} \)
53 \( 1 + 7.89iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 1.65iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + 9.65T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 13.3iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 9.31T + 97T^{2} \)
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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

−9.888252467749468933550164400050, −9.147894224098553478483296824141, −8.333694519213902579866708514614, −7.31547794172270743267699770602, −6.59201299881828062643265550193, −5.95378892975004486847653585040, −4.74820959246706365763185651217, −3.82810363846197742029265266128, −3.00573779890433975733097045612, −1.97952953842009477607474181692, 0.39582559537767615330989319349, 1.57772741526845491161086173725, 2.66954469873067575026387657305, 4.18755519772350732052040507493, 4.96531371720447236051604204426, 5.53256636586702657274805994317, 6.86219544879729193268236531421, 7.33888447006007766954695804298, 8.570004370159834607681382690710, 8.816547576376541846780180407029

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