Properties

Label 2-768-8.5-c5-0-35
Degree $2$
Conductor $768$
Sign $0.707 + 0.707i$
Analytic cond. $123.174$
Root an. cond. $11.0984$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9i·3-s + 66i·5-s − 176·7-s − 81·9-s + 60i·11-s − 658i·13-s + 594·15-s − 414·17-s + 956i·19-s + 1.58e3i·21-s − 600·23-s − 1.23e3·25-s + 729i·27-s + 5.57e3i·29-s − 3.59e3·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.18i·5-s − 1.35·7-s − 0.333·9-s + 0.149i·11-s − 1.07i·13-s + 0.681·15-s − 0.347·17-s + 0.607i·19-s + 0.783i·21-s − 0.236·23-s − 0.393·25-s + 0.192i·27-s + 1.23i·29-s − 0.671·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(123.174\)
Root analytic conductor: \(11.0984\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :5/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9210070900\)
\(L(\frac12)\) \(\approx\) \(0.9210070900\)
\(L(\frac{7}{2})\) 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 \( 1 + 9iT \)
good5 \( 1 - 66iT - 3.12e3T^{2} \)
7 \( 1 + 176T + 1.68e4T^{2} \)
11 \( 1 - 60iT - 1.61e5T^{2} \)
13 \( 1 + 658iT - 3.71e5T^{2} \)
17 \( 1 + 414T + 1.41e6T^{2} \)
19 \( 1 - 956iT - 2.47e6T^{2} \)
23 \( 1 + 600T + 6.43e6T^{2} \)
29 \( 1 - 5.57e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.59e3T + 2.86e7T^{2} \)
37 \( 1 - 8.45e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.91e4T + 1.15e8T^{2} \)
43 \( 1 + 1.33e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.96e4T + 2.29e8T^{2} \)
53 \( 1 - 3.12e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.63e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.10e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.68e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.12e3T + 1.80e9T^{2} \)
73 \( 1 - 2.55e4T + 2.07e9T^{2} \)
79 \( 1 - 7.44e4T + 3.07e9T^{2} \)
83 \( 1 + 6.46e3iT - 3.93e9T^{2} \)
89 \( 1 - 3.27e4T + 5.58e9T^{2} \)
97 \( 1 - 1.66e5T + 8.58e9T^{2} \)
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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

−9.627956951777188351383610021364, −8.515517746779231346042023018420, −7.54579789605196459641968534559, −6.72944144106114820850657613817, −6.28904430562815054720158504414, −5.19383797929831271627884937579, −3.40733619228593721953675994441, −3.13640398197468394010441647131, −1.86630436128317546232919389117, −0.30853220453386472607298916348, 0.58378379597705921947381682142, 2.05755516552421735573510736203, 3.37239726725616879032834180772, 4.26172411170301245831151030338, 5.08567583040468785964965990746, 6.14281496370255785309225603206, 6.91194827726228191032372458387, 8.225918206651193673344884081673, 9.090656248938958834233976442172, 9.461271113192129148826399846248

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