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Properties

Label 2-768-24.11-c3-0-15
Degree $2$
Conductor $768$
Sign $0.778 - 0.627i$
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  + (−5.16 − 0.556i)3-s − 10.6·5-s − 7.90i·7-s + (26.3 + 5.75i)9-s + 11.7i·11-s + 30.5i·13-s + (54.8 + 5.90i)15-s − 118. i·17-s − 66.4·19-s + (−4.40 + 40.8i)21-s − 166.·23-s − 12.4·25-s + (−133. − 44.4i)27-s − 111.·29-s − 224. i·31-s + ⋯
L(s)  = 1  + (−0.994 − 0.107i)3-s − 0.948·5-s − 0.426i·7-s + (0.977 + 0.213i)9-s + 0.322i·11-s + 0.652i·13-s + (0.943 + 0.101i)15-s − 1.68i·17-s − 0.802·19-s + (−0.0457 + 0.424i)21-s − 1.50·23-s − 0.0998·25-s + (−0.948 − 0.316i)27-s − 0.717·29-s − 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.778 - 0.627i$
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: \((2,\ 768,\ (\ :3/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5752895769\)
\(L(\frac12)\) \(\approx\) \(0.5752895769\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.16 + 0.556i)T \)
good5 \( 1 + 10.6T + 125T^{2} \)
7 \( 1 + 7.90iT - 343T^{2} \)
11 \( 1 - 11.7iT - 1.33e3T^{2} \)
13 \( 1 - 30.5iT - 2.19e3T^{2} \)
17 \( 1 + 118. iT - 4.91e3T^{2} \)
19 \( 1 + 66.4T + 6.85e3T^{2} \)
23 \( 1 + 166.T + 1.21e4T^{2} \)
29 \( 1 + 111.T + 2.43e4T^{2} \)
31 \( 1 + 224. iT - 2.97e4T^{2} \)
37 \( 1 - 70.6iT - 5.06e4T^{2} \)
41 \( 1 - 247. iT - 6.89e4T^{2} \)
43 \( 1 + 98.7T + 7.95e4T^{2} \)
47 \( 1 - 189.T + 1.03e5T^{2} \)
53 \( 1 - 529.T + 1.48e5T^{2} \)
59 \( 1 + 811. iT - 2.05e5T^{2} \)
61 \( 1 - 833. iT - 2.26e5T^{2} \)
67 \( 1 - 2.83T + 3.00e5T^{2} \)
71 \( 1 + 796.T + 3.57e5T^{2} \)
73 \( 1 - 875.T + 3.89e5T^{2} \)
79 \( 1 - 656. iT - 4.93e5T^{2} \)
83 \( 1 + 237. iT - 5.71e5T^{2} \)
89 \( 1 - 868. iT - 7.04e5T^{2} \)
97 \( 1 - 755.T + 9.12e5T^{2} \)
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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

−10.09279978938309169081230803124, −9.368239633129948071202030208043, −8.026990823970282953766566655917, −7.35784839239667680354872860712, −6.62868042893617238417852791944, −5.57378451522294488768896209644, −4.43999101056369252539314263205, −3.94575031859547954585141022886, −2.17358645883827702028693090653, −0.61730864749633121338514714538, 0.31515247308560671468474320076, 1.86664423341931714760654753508, 3.63948325500034147373399760008, 4.25635758151870157709075024762, 5.57070461858143352167453514169, 6.09814138952845831833012158117, 7.22233877463051354633925011605, 8.115958325388569783033269854652, 8.840133323030402114188215244066, 10.24209732009265469954920180441

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