Properties

Label 2-384-16.13-c1-0-5
Degree $2$
Conductor $384$
Sign $0.872 + 0.489i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.334 − 0.334i)5-s − 4.55i·7-s + 1.00i·9-s + (2.47 − 2.47i)11-s + (0.0594 + 0.0594i)13-s + 0.473·15-s + 3.61·17-s + (−2.55 − 2.55i)19-s + (3.22 − 3.22i)21-s + 2.82i·23-s + 4.77i·25-s + (−0.707 + 0.707i)27-s + (5.16 + 5.16i)29-s − 0.557·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.149 − 0.149i)5-s − 1.72i·7-s + 0.333i·9-s + (0.745 − 0.745i)11-s + (0.0164 + 0.0164i)13-s + 0.122·15-s + 0.877·17-s + (−0.586 − 0.586i)19-s + (0.703 − 0.703i)21-s + 0.589i·23-s + 0.955i·25-s + (−0.136 + 0.136i)27-s + (0.958 + 0.958i)29-s − 0.100·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56350 - 0.408721i\)
\(L(\frac12)\) \(\approx\) \(1.56350 - 0.408721i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (-0.334 + 0.334i)T - 5iT^{2} \)
7 \( 1 + 4.55iT - 7T^{2} \)
11 \( 1 + (-2.47 + 2.47i)T - 11iT^{2} \)
13 \( 1 + (-0.0594 - 0.0594i)T + 13iT^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 + (2.55 + 2.55i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (-5.16 - 5.16i)T + 29iT^{2} \)
31 \( 1 + 0.557T + 31T^{2} \)
37 \( 1 + (4.38 - 4.38i)T - 37iT^{2} \)
41 \( 1 + 9.27iT - 41T^{2} \)
43 \( 1 + (-1.61 + 1.61i)T - 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-0.493 + 0.493i)T - 53iT^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + (2.72 + 2.72i)T + 61iT^{2} \)
67 \( 1 + (3.77 + 3.77i)T + 67iT^{2} \)
71 \( 1 - 9.11iT - 71T^{2} \)
73 \( 1 + 0.541iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95622444051876198825869411093, −10.45218349938692226375152381503, −9.455655516276754496600053485835, −8.583496898322148199759175567457, −7.49643283851883970303424838483, −6.69148333313821998587039406224, −5.27656954579541939383551991246, −4.07705397765460023138055660320, −3.30688755352605506775610469418, −1.20603024807497834306164677842, 1.89020047741944756437274000865, 2.91289612241523218119961145656, 4.47292682848476598746833523159, 5.85744070877236210604722731863, 6.51108940158708060113879518032, 7.85410705081533497271810187617, 8.651375874094775233556693767952, 9.453482378376616199073254043895, 10.34113250303791978491275644085, 11.84992528163165615995575595763

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