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Properties

Label 2-768-24.5-c2-0-2
Degree $2$
Conductor $768$
Sign $-0.985 + 0.169i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 + 1.73i)3-s + 2.82·5-s − 10.3·7-s + (2.99 + 8.48i)9-s − 14.6·11-s − 6i·13-s + (6.92 + 4.89i)15-s + 22.6i·17-s − 10.3i·19-s + (−25.4 − 18i)21-s − 29.3i·23-s − 17·25-s + (−7.34 + 25.9i)27-s − 31.1·29-s − 31.1·31-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)3-s + 0.565·5-s − 1.48·7-s + (0.333 + 0.942i)9-s − 1.33·11-s − 0.461i·13-s + (0.461 + 0.326i)15-s + 1.33i·17-s − 0.546i·19-s + (−1.21 − 0.857i)21-s − 1.27i·23-s − 0.680·25-s + (−0.272 + 0.962i)27-s − 1.07·29-s − 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.985 + 0.169i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4318383604\)
\(L(\frac12)\) \(\approx\) \(0.4318383604\)
\(L(2)\) 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 \( 1 + (-2.44 - 1.73i)T \)
good5 \( 1 - 2.82T + 25T^{2} \)
7 \( 1 + 10.3T + 49T^{2} \)
11 \( 1 + 14.6T + 121T^{2} \)
13 \( 1 + 6iT - 169T^{2} \)
17 \( 1 - 22.6iT - 289T^{2} \)
19 \( 1 + 10.3iT - 361T^{2} \)
23 \( 1 + 29.3iT - 529T^{2} \)
29 \( 1 + 31.1T + 841T^{2} \)
31 \( 1 + 31.1T + 961T^{2} \)
37 \( 1 - 38iT - 1.36e3T^{2} \)
41 \( 1 - 5.65iT - 1.68e3T^{2} \)
43 \( 1 - 10.3iT - 1.84e3T^{2} \)
47 \( 1 + 58.7iT - 2.20e3T^{2} \)
53 \( 1 - 14.1T + 2.80e3T^{2} \)
59 \( 1 + 14.6T + 3.48e3T^{2} \)
61 \( 1 + 22iT - 3.72e3T^{2} \)
67 \( 1 - 114. iT - 4.48e3T^{2} \)
71 \( 1 - 29.3iT - 5.04e3T^{2} \)
73 \( 1 - 30T + 5.32e3T^{2} \)
79 \( 1 + 31.1T + 6.24e3T^{2} \)
83 \( 1 + 73.4T + 6.88e3T^{2} \)
89 \( 1 - 5.65iT - 7.92e3T^{2} \)
97 \( 1 - 90T + 9.40e3T^{2} \)
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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

−10.18179774265897435094955846171, −9.941664859462614892413681567546, −8.921802594716409495136457190106, −8.189720689495339371580129340228, −7.17992688026881699455566001127, −6.09152894042761829643498960743, −5.26128147126783968281720318087, −3.96180888093498413793600099780, −3.04113696329850970147931218443, −2.17288242567227121422561621114, 0.11726133449850119240940581440, 1.94142990280540867928825417132, 2.91369737425859011357488861201, 3.75730725928778319106569187902, 5.41422530944410619995353252021, 6.17942217857084051145054430832, 7.28670219571116502119988305448, 7.68746520985935758395087152041, 9.172552986958887181106576450205, 9.429581045636624791339653647041

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