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Properties

Label 2-768-48.5-c0-0-2
Degree $2$
Conductor $768$
Sign $0.923 + 0.382i$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
Motivic weight $0$
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 + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s − 1.41·31-s + (−1 − i)37-s − 1.41i·39-s − 1.00·49-s + (−1 + i)61-s + 1.41·63-s + (−1.41 + 1.41i)67-s + (0.707 + 0.707i)75-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + 1.41i·7-s − 1.00i·9-s + (1 − i)13-s + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s − 1.41·31-s + (−1 − i)37-s − 1.41i·39-s − 1.00·49-s + (−1 + i)61-s + 1.41·63-s + (−1.41 + 1.41i)67-s + (0.707 + 0.707i)75-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216303911\)
\(L(\frac12)\) \(\approx\) \(1.216303911\)
\(L(1)\) 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 \( 1 - iT^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.48517391414318431256773648419, −9.180718170801788398644967488632, −8.874579815573019611835329710387, −7.991859654604900860551708282025, −7.14117594131401513683028128746, −5.97765466754364041880612449728, −5.43626017265676189353926315350, −3.70082047242043809456201066502, −2.81946286797672009368096705447, −1.66274092463359388474131334946, 1.73880738035066004212755957453, 3.36019090097599852242217091277, 4.06236654040940025278172943283, 4.87568833691894492966308118606, 6.31594235116094141169306955317, 7.23657877521383497767263340020, 8.100283771869481606588448258401, 8.945645276797791331027864148445, 9.721559713136915931993482560824, 10.64940554845876915518082276543

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