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Properties

Label 2-768-48.29-c0-0-1
Degree $2$
Conductor $768$
Sign $0.382 - 0.923i$
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

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.156811699\)
\(L(\frac12)\) \(\approx\) \(1.156811699\)
\(L(1)\) 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

−10.40570219871736071785993301622, −9.821578329258810840410492178601, −8.978429065104094722462945065131, −8.291823196938352548043833194139, −7.52753259997669659053556414330, −6.09232026584850991134051100787, −5.25414833489590449404043786749, −4.37309667551536545429944692828, −2.94589771446165780746346580217, −2.36187048610256578754138761460, 1.30094034148794529786059142175, 2.67488760753414508913958952736, 3.86446329139798759701981249665, 4.73378768843682497981537909826, 6.31510370080201837787356276565, 7.09799727856111460127818937676, 7.61140859064140470473140862967, 8.567830734604247703833140952229, 9.589024008976271603437400991341, 10.13464452871802418599816595282

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