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Properties

Label 2-768-16.5-c1-0-13
Degree $2$
Conductor $768$
Sign $-0.923 - 0.382i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s + (−2 − 2i)5-s − 4.24i·7-s − 1.00i·9-s + (2.82 + 2.82i)11-s + (−3 + 3i)13-s + 2.82·15-s − 6·17-s + (−1.41 + 1.41i)19-s + (3 + 3i)21-s + 2.82i·23-s + 3i·25-s + (0.707 + 0.707i)27-s + (−4 + 4i)29-s + 4.24·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.894 − 0.894i)5-s − 1.60i·7-s − 0.333i·9-s + (0.852 + 0.852i)11-s + (−0.832 + 0.832i)13-s + 0.730·15-s − 1.45·17-s + (−0.324 + 0.324i)19-s + (0.654 + 0.654i)21-s + 0.589i·23-s + 0.600i·25-s + (0.136 + 0.136i)27-s + (−0.742 + 0.742i)29-s + 0.762·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 2i)T + 5iT^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + (1.41 - 1.41i)T - 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (4 - 4i)T - 29iT^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + (4.24 + 4.24i)T + 43iT^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + (4 + 4i)T + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (3 - 3i)T - 61iT^{2} \)
67 \( 1 + (-2.82 + 2.82i)T - 67iT^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + (11.3 - 11.3i)T - 83iT^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−9.785906623793279037811076601745, −9.155320741707345900899558400300, −8.084728735030911469795895496347, −7.11784251151897158121592936091, −6.61212955784452802837399010896, −4.84626956147050419765203820808, −4.39950211820062654424242074041, −3.75770675270290306082515997269, −1.57030414897983361452482667018, 0, 2.27808580473753400921843480629, 3.14843412171406567916397589057, 4.50695414752409618986653806421, 5.68914249739871708081505360034, 6.44178457033211767961198662404, 7.21926620531824672280521553143, 8.333052827640165494263953521510, 8.852887715292961313027723081612, 10.04141529230404093477285620604

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