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Properties

Label 2-1536-16.13-c1-0-6
Degree $2$
Conductor $1536$
Sign $0.382 - 0.923i$
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  + (−0.707 − 0.707i)3-s + (1 − i)5-s + 2.82i·7-s + 1.00i·9-s + (−3 − 3i)13-s − 1.41·15-s − 4·17-s + (5.65 + 5.65i)19-s + (2.00 − 2.00i)21-s + 5.65i·23-s + 3i·25-s + (0.707 − 0.707i)27-s + (1 + i)29-s + 2.82·31-s + (2.82 + 2.82i)35-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.447 − 0.447i)5-s + 1.06i·7-s + 0.333i·9-s + (−0.832 − 0.832i)13-s − 0.365·15-s − 0.970·17-s + (1.29 + 1.29i)19-s + (0.436 − 0.436i)21-s + 1.17i·23-s + 0.600i·25-s + (0.136 − 0.136i)27-s + (0.185 + 0.185i)29-s + 0.508·31-s + (0.478 + 0.478i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(12.2650\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ 0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.170936591\)
\(L(\frac12)\) \(\approx\) \(1.170936591\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (-5.65 - 5.65i)T + 19iT^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + (-1 - i)T + 29iT^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + (-8.48 + 8.48i)T - 59iT^{2} \)
61 \( 1 + (-1 - i)T + 61iT^{2} \)
67 \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−9.669409235888371071715203667490, −8.804680395600725659443664054282, −8.006492107913185384028186202357, −7.22552981662763402954714374037, −6.19699671260179714881489182902, −5.36876708288940488624598345060, −5.09041130433295868046681319893, −3.49412075972818143864198113107, −2.38621698420814244547139766451, −1.34108666950116522371435134210, 0.49928534871251854727521009319, 2.17161850269381372550967355743, 3.25624638811316762299229158733, 4.55717078881762408938981483721, 4.80677801800913194815602002512, 6.26268778544617292771236439215, 6.80079657530415456894136087300, 7.46251293775756637911281149790, 8.658348783408074030126854707627, 9.521085017801774548370918187328

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