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Properties

Label 2-768-24.5-c2-0-14
Degree $2$
Conductor $768$
Sign $0.169 - 0.985i$
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

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.169 - 0.985i$
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: \((2,\ 768,\ (\ :1),\ 0.169 - 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7139144528\)
\(L(\frac12)\) \(\approx\) \(0.7139144528\)
\(L(2)\) 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

−10.16328343068894191849351185754, −9.524289538776825867742937575844, −9.006958053015805729714012664950, −7.47089452865526022783105672209, −6.69391349806999598012824555654, −5.99277147120950882478915486306, −4.88380448330038075575022462893, −3.81261319620451432218785841852, −3.12044220026226644210272365448, −0.837020041185984528994816062138, 0.39669365840226956002399520981, 1.89949099977462808961943591563, 3.57046650509870409998582967347, 4.29713670996541326291983826841, 5.83361749821617867045342725699, 6.46302000758199172256421701381, 7.00999418136669649561285617521, 8.163633658272952843911025474622, 9.089210800614091683884224632810, 10.07057256821356211204972675501

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