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Properties

Label 2-768-12.11-c3-0-65
Degree $2$
Conductor $768$
Sign $-0.841 + 0.540i$
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  + (−2.80 − 4.37i)3-s − 13.1i·5-s + 26.9i·7-s + (−11.2 + 24.5i)9-s + 21.9·11-s + 10.0·13-s + (−57.5 + 36.9i)15-s − 6.09i·17-s − 40.1i·19-s + (117. − 75.6i)21-s − 9.80·23-s − 48.4·25-s + (138. − 19.8i)27-s − 164. i·29-s − 47.0i·31-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)3-s − 1.17i·5-s + 1.45i·7-s + (−0.416 + 0.909i)9-s + 0.602·11-s + 0.214·13-s + (−0.991 + 0.636i)15-s − 0.0869i·17-s − 0.485i·19-s + (1.22 − 0.786i)21-s − 0.0889·23-s − 0.387·25-s + (0.989 − 0.141i)27-s − 1.05i·29-s − 0.272i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.841 + 0.540i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.089261944\)
\(L(\frac12)\) \(\approx\) \(1.089261944\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.80 + 4.37i)T \)
good5 \( 1 + 13.1iT - 125T^{2} \)
7 \( 1 - 26.9iT - 343T^{2} \)
11 \( 1 - 21.9T + 1.33e3T^{2} \)
13 \( 1 - 10.0T + 2.19e3T^{2} \)
17 \( 1 + 6.09iT - 4.91e3T^{2} \)
19 \( 1 + 40.1iT - 6.85e3T^{2} \)
23 \( 1 + 9.80T + 1.21e4T^{2} \)
29 \( 1 + 164. iT - 2.43e4T^{2} \)
31 \( 1 + 47.0iT - 2.97e4T^{2} \)
37 \( 1 + 205.T + 5.06e4T^{2} \)
41 \( 1 + 419. iT - 6.89e4T^{2} \)
43 \( 1 + 205. iT - 7.95e4T^{2} \)
47 \( 1 - 566.T + 1.03e5T^{2} \)
53 \( 1 - 342. iT - 1.48e5T^{2} \)
59 \( 1 - 3.70T + 2.05e5T^{2} \)
61 \( 1 + 717.T + 2.26e5T^{2} \)
67 \( 1 + 238. iT - 3.00e5T^{2} \)
71 \( 1 - 517.T + 3.57e5T^{2} \)
73 \( 1 + 984.T + 3.89e5T^{2} \)
79 \( 1 - 329. iT - 4.93e5T^{2} \)
83 \( 1 + 625.T + 5.71e5T^{2} \)
89 \( 1 - 238. iT - 7.04e5T^{2} \)
97 \( 1 + 1.00e3T + 9.12e5T^{2} \)
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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

−9.135713078825032123072659426631, −8.871333748804311770104498539286, −7.956111005733090983150998553044, −6.88291618553819571729748951127, −5.81800442266563904156178650282, −5.39632649425624813622737425596, −4.28846582236783659648862556942, −2.56713893555488723019022311193, −1.56413745422656271919421389573, −0.35276721974031741346572582080, 1.19090681722754319208214306478, 3.12920708582353944893227764981, 3.81786905696304916798267674763, 4.70192749426134786280640900828, 6.01031569159696131419082598985, 6.75962746959069066042639014419, 7.43148343373538537036821930935, 8.698984872598358536306159969352, 9.779434238630264373443680297729, 10.41114883136018962930712722519

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