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Properties

Label 2-768-96.35-c1-0-24
Degree $2$
Conductor $768$
Sign $-0.672 + 0.740i$
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  + (−1.25 + 1.19i)3-s + (2.18 − 0.906i)5-s + (−1.93 − 1.93i)7-s + (0.131 − 2.99i)9-s + (−1.42 + 0.590i)11-s + (0.110 − 0.266i)13-s + (−1.65 + 3.75i)15-s − 6.17·17-s + (−7.34 − 3.04i)19-s + (4.73 + 0.103i)21-s + (1.85 + 1.85i)23-s + (0.430 − 0.430i)25-s + (3.42 + 3.90i)27-s + (−2.11 + 5.10i)29-s − 3.42i·31-s + ⋯
L(s)  = 1  + (−0.722 + 0.691i)3-s + (0.978 − 0.405i)5-s + (−0.730 − 0.730i)7-s + (0.0439 − 0.999i)9-s + (−0.429 + 0.177i)11-s + (0.0306 − 0.0739i)13-s + (−0.426 + 0.969i)15-s − 1.49·17-s + (−1.68 − 0.697i)19-s + (1.03 + 0.0226i)21-s + (0.386 + 0.386i)23-s + (0.0861 − 0.0861i)25-s + (0.659 + 0.752i)27-s + (−0.392 + 0.948i)29-s − 0.614i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.672 + 0.740i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.672 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171831 - 0.388386i\)
\(L(\frac12)\) \(\approx\) \(0.171831 - 0.388386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 - 1.19i)T \)
good5 \( 1 + (-2.18 + 0.906i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.93 + 1.93i)T + 7iT^{2} \)
11 \( 1 + (1.42 - 0.590i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.110 + 0.266i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.17T + 17T^{2} \)
19 \( 1 + (7.34 + 3.04i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.85 - 1.85i)T + 23iT^{2} \)
29 \( 1 + (2.11 - 5.10i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 3.42iT - 31T^{2} \)
37 \( 1 + (2.52 + 6.09i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.753 - 0.753i)T - 41iT^{2} \)
43 \( 1 + (1.57 + 3.79i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.54iT - 47T^{2} \)
53 \( 1 + (5.12 + 12.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.08 - 7.45i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-4.28 - 1.77i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-0.531 + 1.28i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-8.72 + 8.72i)T - 71iT^{2} \)
73 \( 1 + (2.73 + 2.73i)T + 73iT^{2} \)
79 \( 1 + 2.76T + 79T^{2} \)
83 \( 1 + (2.53 - 6.11i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.14 - 4.14i)T + 89iT^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−10.08811266729420483562786835805, −9.289138809412237567262829515195, −8.678115587357187939661907747903, −7.02504644518899058983835398315, −6.46967320498843599992933204021, −5.48744994544657879039747001345, −4.63898296345845455551592055765, −3.68252964920529177576522118892, −2.11676047661180096819860968159, −0.21313402281341012299862379217, 1.94729723060687989141469365987, 2.68721390325631846765740849340, 4.47000059542879786003901524201, 5.63460850422824182185199307402, 6.36832278244326847666328568242, 6.68825710765261638510896470545, 8.095289807231327934128183136525, 8.914944643586794699795646104300, 9.955934136829711753320791396207, 10.64979220620082676315844134150

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