↑

Properties

Label 2-384-128.53-c1-0-3
Degree $2$
Conductor $384$
Sign $0.906 - 0.421i$
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  + (−1.28 − 0.596i)2-s + (−0.0980 − 0.995i)3-s + (1.28 + 1.53i)4-s + (−0.434 − 0.232i)5-s + (−0.468 + 1.33i)6-s + (−0.519 + 2.60i)7-s + (−0.736 − 2.73i)8-s + (−0.980 + 0.195i)9-s + (0.418 + 0.556i)10-s + (0.916 + 1.11i)11-s + (1.39 − 1.43i)12-s + (1.88 + 3.52i)13-s + (2.22 − 3.03i)14-s + (−0.188 + 0.455i)15-s + (−0.685 + 3.94i)16-s + (0.865 + 2.08i)17-s + ⋯
L(s)  = 1  + (−0.906 − 0.422i)2-s + (−0.0565 − 0.574i)3-s + (0.643 + 0.765i)4-s + (−0.194 − 0.103i)5-s + (−0.191 + 0.544i)6-s + (−0.196 + 0.986i)7-s + (−0.260 − 0.965i)8-s + (−0.326 + 0.0650i)9-s + (0.132 + 0.176i)10-s + (0.276 + 0.336i)11-s + (0.403 − 0.413i)12-s + (0.523 + 0.978i)13-s + (0.594 − 0.811i)14-s + (−0.0486 + 0.117i)15-s + (−0.171 + 0.985i)16-s + (0.209 + 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.906 - 0.421i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.906 - 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755421 + 0.166912i\)
\(L(\frac12)\) \(\approx\) \(0.755421 + 0.166912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.596i)T \)
3 \( 1 + (0.0980 + 0.995i)T \)
good5 \( 1 + (0.434 + 0.232i)T + (2.77 + 4.15i)T^{2} \)
7 \( 1 + (0.519 - 2.60i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.916 - 1.11i)T + (-2.14 + 10.7i)T^{2} \)
13 \( 1 + (-1.88 - 3.52i)T + (-7.22 + 10.8i)T^{2} \)
17 \( 1 + (-0.865 - 2.08i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (0.820 - 2.70i)T + (-15.7 - 10.5i)T^{2} \)
23 \( 1 + (1.61 + 1.07i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-6.40 - 5.25i)T + (5.65 + 28.4i)T^{2} \)
31 \( 1 + (-4.65 - 4.65i)T + 31iT^{2} \)
37 \( 1 + (-6.97 + 2.11i)T + (30.7 - 20.5i)T^{2} \)
41 \( 1 + (-5.57 + 8.34i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (0.656 - 6.66i)T + (-42.1 - 8.38i)T^{2} \)
47 \( 1 + (7.80 - 3.23i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (1.88 - 1.55i)T + (10.3 - 51.9i)T^{2} \)
59 \( 1 + (1.29 - 2.42i)T + (-32.7 - 49.0i)T^{2} \)
61 \( 1 + (-5.00 + 0.493i)T + (59.8 - 11.9i)T^{2} \)
67 \( 1 + (5.17 - 0.509i)T + (65.7 - 13.0i)T^{2} \)
71 \( 1 + (-1.15 - 0.228i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (1.75 + 8.80i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (0.912 + 0.378i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.02 + 1.82i)T + (69.0 + 46.1i)T^{2} \)
89 \( 1 + (15.5 - 10.3i)T + (34.0 - 82.2i)T^{2} \)
97 \( 1 + (-4.43 - 4.43i)T + 97iT^{2} \)
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

−11.49019831090589820425975543633, −10.47676546602652009739189991146, −9.452928077616428427661547236539, −8.627271078380499759404541302943, −7.975359796526419754615772769855, −6.71611795800680902261089253494, −6.04440334994052476621697030726, −4.21264496508934714424460835066, −2.74163242579487972447511036475, −1.52597656498385246535700831215, 0.74937988433260736527084423607, 2.96263086723384467514307435195, 4.33957890217940560026576889829, 5.69165713937736073244514872997, 6.62817355015312076829037466034, 7.72619480135927290920591957674, 8.409579785249320425749232519446, 9.668448230789390332600238471132, 10.10315093847408792874729305303, 11.12630291609404563541773649883

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