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Properties

Label 2-768-16.11-c2-0-11
Degree $2$
Conductor $768$
Sign $-0.130 - 0.991i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (6.45 + 6.45i)5-s − 8.74·7-s + 2.99i·9-s + (9.12 − 9.12i)11-s + (−9.18 + 9.18i)13-s − 15.8i·15-s + 18.9·17-s + (2.79 + 2.79i)19-s + (10.7 + 10.7i)21-s + 2.27·23-s + 58.2i·25-s + (3.67 − 3.67i)27-s + (−10.8 + 10.8i)29-s + 50.9i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (1.29 + 1.29i)5-s − 1.24·7-s + 0.333i·9-s + (0.829 − 0.829i)11-s + (−0.706 + 0.706i)13-s − 1.05i·15-s + 1.11·17-s + (0.147 + 0.147i)19-s + (0.510 + 0.510i)21-s + 0.0988·23-s + 2.33i·25-s + (0.136 − 0.136i)27-s + (−0.375 + 0.375i)29-s + 1.64i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.409615791\)
\(L(\frac12)\) \(\approx\) \(1.409615791\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (-6.45 - 6.45i)T + 25iT^{2} \)
7 \( 1 + 8.74T + 49T^{2} \)
11 \( 1 + (-9.12 + 9.12i)T - 121iT^{2} \)
13 \( 1 + (9.18 - 9.18i)T - 169iT^{2} \)
17 \( 1 - 18.9T + 289T^{2} \)
19 \( 1 + (-2.79 - 2.79i)T + 361iT^{2} \)
23 \( 1 - 2.27T + 529T^{2} \)
29 \( 1 + (10.8 - 10.8i)T - 841iT^{2} \)
31 \( 1 - 50.9iT - 961T^{2} \)
37 \( 1 + (47.2 + 47.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 27.6iT - 1.68e3T^{2} \)
43 \( 1 + (35.9 - 35.9i)T - 1.84e3iT^{2} \)
47 \( 1 - 61.0iT - 2.20e3T^{2} \)
53 \( 1 + (-53.7 - 53.7i)T + 2.80e3iT^{2} \)
59 \( 1 + (20.5 - 20.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (12.6 - 12.6i)T - 3.72e3iT^{2} \)
67 \( 1 + (43.7 + 43.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 20.5T + 5.04e3T^{2} \)
73 \( 1 - 107. iT - 5.32e3T^{2} \)
79 \( 1 - 49.9iT - 6.24e3T^{2} \)
83 \( 1 + (15.0 + 15.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 62.0iT - 7.92e3T^{2} \)
97 \( 1 + 22.7T + 9.40e3T^{2} \)
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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

−10.33532223461159793760810573007, −9.616065964905401341086655980787, −8.953066781728271932387544275566, −7.29745288305310398969303017657, −6.77496580675091382301772400105, −6.09659272942487218759299541787, −5.40918614933451922556597577383, −3.55272677825926958984606377762, −2.79247563526266049823915337947, −1.47552871102835171443116187090, 0.51003893739363163298912569388, 1.89657299181410554396541371469, 3.37923387272446937203760680054, 4.64292334552947289743745953502, 5.44548455144868396807545518954, 6.13270326599961641601246071159, 7.09116684484984603460779673264, 8.435759764463776571702041026002, 9.418203291123848010708509547923, 9.863945474743451453825465956315

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