Properties

Label 2-768-16.5-c1-0-0
Degree $2$
Conductor $768$
Sign $-0.382 - 0.923i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
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 + 1.41i·7-s − 1.00i·9-s + (−1 + i)13-s + 6·17-s + (−4.24 + 4.24i)19-s + (−1.00 − 1.00i)21-s + 8.48i·23-s − 5i·25-s + (0.707 + 0.707i)27-s + (−6 + 6i)29-s − 1.41·31-s + (−5 − 5i)37-s − 1.41i·39-s + 6i·41-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + 0.534i·7-s − 0.333i·9-s + (−0.277 + 0.277i)13-s + 1.45·17-s + (−0.973 + 0.973i)19-s + (−0.218 − 0.218i)21-s + 1.76i·23-s i·25-s + (0.136 + 0.136i)27-s + (−1.11 + 1.11i)29-s − 0.254·31-s + (−0.821 − 0.821i)37-s − 0.226i·39-s + 0.937i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561004 + 0.839602i\)
\(L(\frac12)\) \(\approx\) \(0.561004 + 0.839602i\)
\(L(\frac{3}{2})\) 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 \( 1 + 5iT^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (4.24 - 4.24i)T - 19iT^{2} \)
23 \( 1 - 8.48iT - 23T^{2} \)
29 \( 1 + (6 - 6i)T - 29iT^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 + (-6 - 6i)T + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (5 - 5i)T - 61iT^{2} \)
67 \( 1 + (2.82 - 2.82i)T - 67iT^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 1.41T + 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−10.52110694422570772889004335844, −9.782601191833913671875059868283, −9.036757784577249241503103294656, −8.020857896992264591328525490687, −7.14773171218612921671672005667, −5.89082707323611921294367441610, −5.44579491230146236877156114193, −4.19071049242225518080767056514, −3.20942371647458972517212775952, −1.65570692786830293621000380784, 0.54199357997023415177221832529, 2.13967858469280935062734473594, 3.52307585605867997350083267035, 4.70005743981252553254052781398, 5.62739466846925634789877862234, 6.63279219802025869504953037582, 7.39194044083058310715094072917, 8.209554747461376406928244404405, 9.222051040982898383324703723179, 10.27406048793311052029188765143

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