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Properties

Label 2-768-32.13-c1-0-14
Degree $2$
Conductor $768$
Sign $0.0241 + 0.999i$
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  + (0.382 − 0.923i)3-s + (1.46 − 0.605i)5-s + (3.54 − 3.54i)7-s + (−0.707 − 0.707i)9-s + (−0.471 − 1.13i)11-s + (−4.97 − 2.05i)13-s − 1.58i·15-s − 0.419i·17-s + (−0.721 − 0.298i)19-s + (−1.92 − 4.63i)21-s + (5.76 + 5.76i)23-s + (−1.76 + 1.76i)25-s + (−0.923 + 0.382i)27-s + (−1.26 + 3.05i)29-s − 0.702·31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (0.653 − 0.270i)5-s + (1.34 − 1.34i)7-s + (−0.235 − 0.235i)9-s + (−0.142 − 0.342i)11-s + (−1.37 − 0.570i)13-s − 0.408i·15-s − 0.101i·17-s + (−0.165 − 0.0685i)19-s + (−0.419 − 1.01i)21-s + (1.20 + 1.20i)23-s + (−0.352 + 0.352i)25-s + (−0.177 + 0.0736i)27-s + (−0.234 + 0.566i)29-s − 0.126·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0241 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0241 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.0241 + 0.999i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.0241 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38019 - 1.34729i\)
\(L(\frac12)\) \(\approx\) \(1.38019 - 1.34729i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (-1.46 + 0.605i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.54 + 3.54i)T - 7iT^{2} \)
11 \( 1 + (0.471 + 1.13i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (4.97 + 2.05i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.419iT - 17T^{2} \)
19 \( 1 + (0.721 + 0.298i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.76 - 5.76i)T + 23iT^{2} \)
29 \( 1 + (1.26 - 3.05i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 0.702T + 31T^{2} \)
37 \( 1 + (1.86 - 0.773i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.76 - 1.76i)T + 41iT^{2} \)
43 \( 1 + (-1.70 - 4.12i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 9.64iT - 47T^{2} \)
53 \( 1 + (-0.729 - 1.76i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-9.04 + 3.74i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.0348 - 0.0842i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.84 + 4.44i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-4.81 + 4.81i)T - 71iT^{2} \)
73 \( 1 + (-4.70 - 4.70i)T + 73iT^{2} \)
79 \( 1 - 2.83iT - 79T^{2} \)
83 \( 1 + (-8.15 - 3.37i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (5.34 - 5.34i)T - 89iT^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−10.11921428602942409882363285204, −9.286852227400028994210084827004, −8.204814418512651456045699134113, −7.51039625855342907634535542431, −6.93027996477944573626395056686, −5.41936780356518531927193388187, −4.90299589335425499973009962484, −3.53740772288240574648684800949, −2.10174123251332055852116962854, −0.998523726767266011244041068417, 2.08297272289036443659407063650, 2.58587851783559321931692336784, 4.42379597786281136913658389590, 5.07217676161145096335595199695, 5.91597743662261732602489034252, 7.12344951726520036429801994278, 8.141385682554676211164871365071, 8.931227958640651290490428659363, 9.591630341808975858020639882780, 10.47557797263586249906161853396

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