Properties

Label 2-768-16.3-c2-0-2
Degree $2$
Conductor $768$
Sign $-0.608 - 0.793i$
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 + (2 − 2i)5-s − 5.27·7-s − 2.99i·9-s + (−0.757 − 0.757i)11-s + (0.464 + 0.464i)13-s + 4.89i·15-s + 15.8·17-s + (−3.96 + 3.96i)19-s + (6.46 − 6.46i)21-s − 21.8·23-s + 17i·25-s + (3.67 + 3.67i)27-s + (−14.9 − 14.9i)29-s + 57.2i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.400 − 0.400i)5-s − 0.753·7-s − 0.333i·9-s + (−0.0688 − 0.0688i)11-s + (0.0357 + 0.0357i)13-s + 0.326i·15-s + 0.932·17-s + (−0.208 + 0.208i)19-s + (0.307 − 0.307i)21-s − 0.950·23-s + 0.680i·25-s + (0.136 + 0.136i)27-s + (−0.514 − 0.514i)29-s + 1.84i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7649734904\)
\(L(\frac12)\) \(\approx\) \(0.7649734904\)
\(L(2)\) 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 \( 1 + (-2 + 2i)T - 25iT^{2} \)
7 \( 1 + 5.27T + 49T^{2} \)
11 \( 1 + (0.757 + 0.757i)T + 121iT^{2} \)
13 \( 1 + (-0.464 - 0.464i)T + 169iT^{2} \)
17 \( 1 - 15.8T + 289T^{2} \)
19 \( 1 + (3.96 - 3.96i)T - 361iT^{2} \)
23 \( 1 + 21.8T + 529T^{2} \)
29 \( 1 + (14.9 + 14.9i)T + 841iT^{2} \)
31 \( 1 - 57.2iT - 961T^{2} \)
37 \( 1 + (-14.6 + 14.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 79.5iT - 1.68e3T^{2} \)
43 \( 1 + (15.2 + 15.2i)T + 1.84e3iT^{2} \)
47 \( 1 - 2.27iT - 2.20e3T^{2} \)
53 \( 1 + (-27.2 + 27.2i)T - 2.80e3iT^{2} \)
59 \( 1 + (-50.5 - 50.5i)T + 3.48e3iT^{2} \)
61 \( 1 + (68.8 + 68.8i)T + 3.72e3iT^{2} \)
67 \( 1 + (59.9 - 59.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 82.9T + 5.04e3T^{2} \)
73 \( 1 - 77.8iT - 5.32e3T^{2} \)
79 \( 1 - 18.0iT - 6.24e3T^{2} \)
83 \( 1 + (99.9 - 99.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 74iT - 7.92e3T^{2} \)
97 \( 1 - 49.2T + 9.40e3T^{2} \)
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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

−10.17017864085457321587371021538, −9.772301573429391156813094133610, −8.885578165951087433374827861142, −7.905257695236806769471876142991, −6.77488383534656895278181260376, −5.90161056399133482863774098886, −5.19169199206935851950816508549, −4.01798629296278145922989171448, −3.00586057790073082404246760891, −1.37101478478864225997441498415, 0.28095332027672317348796456317, 1.94342721669344232533983991534, 3.09497548099768981127852555850, 4.30584866255505946688752382085, 5.70690922352787143653322763416, 6.15889414192675409874728829916, 7.16124222526823278764134497687, 7.923673253987360692740436460803, 9.084104230750717196407979359335, 9.996284097425504338729724451684

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