↑

Properties

Label 2-384-128.101-c1-0-15
Degree $2$
Conductor $384$
Sign $0.446 - 0.894i$
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  + (1.19 + 0.755i)2-s + (0.881 − 0.471i)3-s + (0.857 + 1.80i)4-s + (0.324 + 0.396i)5-s + (1.41 + 0.103i)6-s + (−2.62 + 1.75i)7-s + (−0.341 + 2.80i)8-s + (0.555 − 0.831i)9-s + (0.0890 + 0.718i)10-s + (2.66 − 0.809i)11-s + (1.60 + 1.18i)12-s + (3.50 + 2.87i)13-s + (−4.46 + 0.112i)14-s + (0.473 + 0.196i)15-s + (−2.53 + 3.09i)16-s + (−1.24 + 0.514i)17-s + ⋯
L(s)  = 1  + (0.845 + 0.534i)2-s + (0.509 − 0.272i)3-s + (0.428 + 0.903i)4-s + (0.145 + 0.177i)5-s + (0.575 + 0.0421i)6-s + (−0.992 + 0.662i)7-s + (−0.120 + 0.992i)8-s + (0.185 − 0.277i)9-s + (0.0281 + 0.227i)10-s + (0.804 − 0.243i)11-s + (0.464 + 0.343i)12-s + (0.973 + 0.798i)13-s + (−1.19 + 0.0299i)14-s + (0.122 + 0.0506i)15-s + (−0.632 + 0.774i)16-s + (−0.301 + 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.446 - 0.894i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08197 + 1.28836i\)
\(L(\frac12)\) \(\approx\) \(2.08197 + 1.28836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.755i)T \)
3 \( 1 + (-0.881 + 0.471i)T \)
good5 \( 1 + (-0.324 - 0.396i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (2.62 - 1.75i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-2.66 + 0.809i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-3.50 - 2.87i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (1.24 - 0.514i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.508 + 5.16i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (-0.610 + 0.121i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.53 + 5.06i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (4.71 + 4.71i)T + 31iT^{2} \)
37 \( 1 + (0.337 + 0.0332i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (0.532 + 2.67i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (4.98 + 2.66i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (-2.42 - 5.85i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-3.19 - 10.5i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (10.5 - 8.66i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (1.48 + 2.77i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (5.88 + 11.0i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (4.98 + 7.45i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-6.36 - 4.25i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-3.81 + 9.21i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-12.4 + 1.22i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (-12.3 - 2.45i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (4.31 + 4.31i)T + 97iT^{2} \)
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

−11.80726465088447801809203936952, −10.81925962432706882078873813872, −9.171042501514572017787818352499, −8.882570928759017211725390472648, −7.54956972659997407885009421772, −6.33921029106982572069739512610, −6.22523988090503505899846601603, −4.46362214801544550403374400658, −3.42269140370008155292988479275, −2.31641331963299723725309191666, 1.47746986677270387049355725629, 3.31728821969332947459250254220, 3.76382713583253674623070498112, 5.14657035866900396714108742925, 6.29294771571079693766603621632, 7.14874762159738996609566663762, 8.635464121696420418617630915017, 9.613535169650799520918292629332, 10.32530705249393316551952674440, 11.09986252947441547157263552603

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