Properties

Label 2-1536-16.13-c1-0-4
Degree $2$
Conductor $1536$
Sign $-0.923 - 0.382i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-0.923 - 0.382i$
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: \((2,\ 1536,\ (\ :1/2),\ -0.923 - 0.382i)\)

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

−9.589630895468636743739768650988, −8.858463338049207374489019470894, −8.562635953296538788481631946511, −7.37810294229411144934309132996, −6.60533721879017309981301527491, −5.74941942957228156403274914040, −4.65128054919586659214019225341, −3.87330137131715963396298802163, −2.82740154341321765283510203429, −1.92860737206447420557457776992, 0.42234843093892868677218902358, 1.66951985553575565823282547879, 3.03799731554637758135619553233, 4.08532292656413165095716198861, 4.60142970613711549972779290239, 6.12893948418251827249722409462, 6.60735782479540872737213519337, 7.75260811035420465010281710778, 8.250160355761547726068371573570, 8.795841347477245402550692365460

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