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Properties

Label 2-768-12.11-c1-0-10
Degree $2$
Conductor $768$
Sign $1$
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  − 1.73i·3-s + 4i·7-s − 2.99·9-s + 6.92·13-s + 3.46i·19-s + 6.92·21-s + 5·25-s + 5.19i·27-s + 4i·31-s + 6.92·37-s − 11.9i·39-s − 10.3i·43-s − 9·49-s + 5.99·57-s + 6.92·61-s + ⋯
L(s)  = 1  − 0.999i·3-s + 1.51i·7-s − 0.999·9-s + 1.92·13-s + 0.794i·19-s + 1.51·21-s + 25-s + 0.999i·27-s + 0.718i·31-s + 1.13·37-s − 1.92i·39-s − 1.58i·43-s − 1.28·49-s + 0.794·57-s + 0.887·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $1$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57067\)
\(L(\frac12)\) \(\approx\) \(1.57067\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.92T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6.92T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.48500059087346643667647241012, −9.033172505212252877586664079770, −8.663285819529977919185795192280, −7.897838459503657549846363798341, −6.64905696871188257101548461427, −6.00420247134769971924398583236, −5.29609427236537451164399649436, −3.60228006940604190025792427889, −2.50955281868900227639178013999, −1.33497880206346318025289810884, 0.961431116756912143762059439283, 3.04524455025905021556332456747, 3.98468915010607806036366835180, 4.60861140518971311200195764729, 5.89359848084046176792417654229, 6.74722578855258813372901593312, 7.88571706082725606057267744158, 8.707081547534969049035240311923, 9.565662713196256120989989225114, 10.43263394701690631820877707255

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