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Properties

Label 2-768-24.11-c3-0-14
Degree $2$
Conductor $768$
Sign $-0.275 - 0.961i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.54 − 2.52i)3-s + 8.01·5-s + 12.6i·7-s + (14.2 + 22.9i)9-s − 37.7i·11-s + 60.0i·13-s + (−36.3 − 20.2i)15-s + 37.0i·17-s + 127.·19-s + (31.7 − 57.2i)21-s − 56.9·23-s − 60.8·25-s + (−7.09 − 140. i)27-s − 220.·29-s + 2.26i·31-s + ⋯
L(s)  = 1  + (−0.874 − 0.485i)3-s + 0.716·5-s + 0.680i·7-s + (0.528 + 0.848i)9-s − 1.03i·11-s + 1.28i·13-s + (−0.626 − 0.347i)15-s + 0.528i·17-s + 1.54·19-s + (0.330 − 0.594i)21-s − 0.516·23-s − 0.486·25-s + (−0.0505 − 0.998i)27-s − 1.41·29-s + 0.0130i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.275 - 0.961i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.275 - 0.961i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9795565187\)
\(L(\frac12)\) \(\approx\) \(0.9795565187\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.54 + 2.52i)T \)
good5 \( 1 - 8.01T + 125T^{2} \)
7 \( 1 - 12.6iT - 343T^{2} \)
11 \( 1 + 37.7iT - 1.33e3T^{2} \)
13 \( 1 - 60.0iT - 2.19e3T^{2} \)
17 \( 1 - 37.0iT - 4.91e3T^{2} \)
19 \( 1 - 127.T + 6.85e3T^{2} \)
23 \( 1 + 56.9T + 1.21e4T^{2} \)
29 \( 1 + 220.T + 2.43e4T^{2} \)
31 \( 1 - 2.26iT - 2.97e4T^{2} \)
37 \( 1 - 166. iT - 5.06e4T^{2} \)
41 \( 1 + 154. iT - 6.89e4T^{2} \)
43 \( 1 + 53.4T + 7.95e4T^{2} \)
47 \( 1 + 591.T + 1.03e5T^{2} \)
53 \( 1 - 538.T + 1.48e5T^{2} \)
59 \( 1 + 586. iT - 2.05e5T^{2} \)
61 \( 1 - 431. iT - 2.26e5T^{2} \)
67 \( 1 + 175.T + 3.00e5T^{2} \)
71 \( 1 - 29.5T + 3.57e5T^{2} \)
73 \( 1 + 937.T + 3.89e5T^{2} \)
79 \( 1 - 409. iT - 4.93e5T^{2} \)
83 \( 1 - 1.29e3iT - 5.71e5T^{2} \)
89 \( 1 - 921. iT - 7.04e5T^{2} \)
97 \( 1 + 1.64e3T + 9.12e5T^{2} \)
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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

−10.14200532782007064826207739357, −9.428881611020968682250090362223, −8.497118832877402849766808097476, −7.45960026468181462132190680595, −6.46793204452501672372756651862, −5.78580608982659201849031617251, −5.17792586354237610994510993971, −3.75868825427881882692333895000, −2.24609467524522782082291964539, −1.30732652336855644677903290416, 0.30748070090180986942609933053, 1.59330506666670884078741757249, 3.20586361068395503866067378881, 4.32763798905120953630925110699, 5.33546588402661976652288877965, 5.86567442157119346967912966300, 7.12220665428018057071332250979, 7.65697655503676175925630922239, 9.208405507239174941796071596098, 9.977116025791918528058986231427

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