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Properties

Label 2-1536-16.13-c1-0-15
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 + (2.23 − 2.23i)5-s + 4.57i·7-s + 1.00i·9-s + (−1.74 + 1.74i)11-s + (1 + i)13-s + 3.16·15-s + 6.47·17-s + (−1.74 − 1.74i)19-s + (−3.23 + 3.23i)21-s + 5.65i·23-s − 5.00i·25-s + (−0.707 + 0.707i)27-s + (−0.236 − 0.236i)29-s − 10.2·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.999 − 0.999i)5-s + 1.72i·7-s + 0.333i·9-s + (−0.527 + 0.527i)11-s + (0.277 + 0.277i)13-s + 0.816·15-s + 1.56·17-s + (−0.401 − 0.401i)19-s + (−0.706 + 0.706i)21-s + 1.17i·23-s − 1.00i·25-s + (−0.136 + 0.136i)27-s + (−0.0438 − 0.0438i)29-s − 1.83·31-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

\(L(1)\) \(\approx\) \(2.268077815\)
\(L(\frac12)\) \(\approx\) \(2.268077815\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 2.23i)T - 5iT^{2} \)
7 \( 1 - 4.57iT - 7T^{2} \)
11 \( 1 + (1.74 - 1.74i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 + (1.74 + 1.74i)T + 19iT^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + (0.236 + 0.236i)T + 29iT^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + (1.47 - 1.47i)T - 37iT^{2} \)
41 \( 1 - 6.47iT - 41T^{2} \)
43 \( 1 + (-7.40 + 7.40i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (3.76 - 3.76i)T - 53iT^{2} \)
59 \( 1 + (-6.32 + 6.32i)T - 59iT^{2} \)
61 \( 1 + (-7.47 - 7.47i)T + 61iT^{2} \)
67 \( 1 + (-8.48 - 8.48i)T + 67iT^{2} \)
71 \( 1 + 3.49iT - 71T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 - 1.08T + 79T^{2} \)
83 \( 1 + (1.74 + 1.74i)T + 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 - 4.94T + 97T^{2} \)
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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

−9.471832170968212672779793091549, −8.970864343912536260888761101431, −8.306092568985560461932409802929, −7.35175925096649925197841594172, −5.90344676924411063179682766177, −5.49020101141999383551433275130, −4.90153534783816095201352862207, −3.52531488446330335189865268430, −2.38834539746110669562443880735, −1.62364611119692392001815848341, 0.867897769378288760701957879509, 2.14216328638014757530548573511, 3.24794731641857081841506584908, 3.89909434353405739871051793846, 5.37207495034155464662346597983, 6.17988935611815398373417666613, 6.98314212568140964817036748332, 7.58658941391845802947828444119, 8.315324265054083774988463423583, 9.499052530804220182914204687086

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