Properties

Label 2-768-24.11-c1-0-18
Degree $2$
Conductor $768$
Sign $-0.707 + 0.707i$
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  − 1.73·3-s − 3.46i·7-s + 2.99·9-s + 2i·13-s − 3.46·19-s + 5.99i·21-s − 5·25-s − 5.19·27-s − 10.3i·31-s − 10i·37-s − 3.46i·39-s − 10.3·43-s − 4.99·49-s + 5.99·57-s − 14i·61-s + ⋯
L(s)  = 1  − 1.00·3-s − 1.30i·7-s + 0.999·9-s + 0.554i·13-s − 0.794·19-s + 1.30i·21-s − 25-s − 1.00·27-s − 1.86i·31-s − 1.64i·37-s − 0.554i·39-s − 1.58·43-s − 0.714·49-s + 0.794·57-s − 1.79i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.223854 - 0.540433i\)
\(L(\frac12)\) \(\approx\) \(0.223854 - 0.540433i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 14iT - 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 - 17.3iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.09525095594102606300989027275, −9.466453164201414882216056540084, −8.072311566728483239187147642916, −7.25262851361504783896179659909, −6.52019909053280814191114002591, −5.61409552938894761200238742026, −4.40323874543549743438045570322, −3.86836922592150014121603379110, −1.86883732524368011485336623159, −0.33364763152082249397841607729, 1.67924139145711383148454219131, 3.09756281512473601456889499708, 4.55296030389065355570238880970, 5.41006058619133677801391830768, 6.11256316522981164088914181311, 6.95956252353826936079512097918, 8.162147062606900746358060781358, 8.915001357845031941750669566980, 10.00487190366944066530946591970, 10.57869600085920685619729127660

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