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Properties

Label 2-384-128.101-c1-0-29
Degree $2$
Conductor $384$
Sign $-0.338 + 0.940i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.986 − 1.01i)2-s + (0.881 − 0.471i)3-s + (−0.0550 − 1.99i)4-s + (−0.677 − 0.825i)5-s + (0.391 − 1.35i)6-s + (1.41 − 0.942i)7-s + (−2.08 − 1.91i)8-s + (0.555 − 0.831i)9-s + (−1.50 − 0.127i)10-s + (−1.74 + 0.528i)11-s + (−0.991 − 1.73i)12-s + (2.10 + 1.72i)13-s + (0.435 − 2.35i)14-s + (−0.987 − 0.408i)15-s + (−3.99 + 0.220i)16-s + (−1.28 + 0.531i)17-s + ⋯
L(s)  = 1  + (0.697 − 0.716i)2-s + (0.509 − 0.272i)3-s + (−0.0275 − 0.999i)4-s + (−0.303 − 0.369i)5-s + (0.159 − 0.554i)6-s + (0.532 − 0.356i)7-s + (−0.735 − 0.677i)8-s + (0.185 − 0.277i)9-s + (−0.476 − 0.0402i)10-s + (−0.525 + 0.159i)11-s + (−0.286 − 0.501i)12-s + (0.582 + 0.478i)13-s + (0.116 − 0.630i)14-s + (−0.254 − 0.105i)15-s + (−0.998 + 0.0550i)16-s + (−0.311 + 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.338 + 0.940i$
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: \((2,\ 384,\ (\ :1/2),\ -0.338 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24500 - 1.77110i\)
\(L(\frac12)\) \(\approx\) \(1.24500 - 1.77110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.986 + 1.01i)T \)
3 \( 1 + (-0.881 + 0.471i)T \)
good5 \( 1 + (0.677 + 0.825i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-1.41 + 0.942i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (1.74 - 0.528i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (-2.10 - 1.72i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (1.28 - 0.531i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.136 - 1.38i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (-3.76 + 0.749i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (1.26 - 4.16i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-0.429 - 0.429i)T + 31iT^{2} \)
37 \( 1 + (-9.15 - 0.902i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-0.0409 - 0.205i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.656 - 0.350i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (1.66 + 4.01i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-2.99 - 9.86i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (1.91 - 1.57i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-6.69 - 12.5i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (5.87 + 10.9i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (1.22 + 1.83i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-4.58 - 3.06i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-2.27 + 5.49i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (7.02 - 0.691i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (14.5 + 2.89i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-7.28 - 7.28i)T + 97iT^{2} \)
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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

−11.13850017092324574044274160445, −10.40786226298361561353566098163, −9.255350084311444488115264290387, −8.405604168614777756615093075802, −7.28596403164716075860441585977, −6.12279593196783262957449467140, −4.82020850013376936145445942416, −4.01908839830142223610661758730, −2.69073783774904859692863747365, −1.28450904500149154822904803008, 2.58031704336840290658352333519, 3.61900774918983177860402969574, 4.79626196138047377357566560957, 5.72774436141476622855915888569, 6.95242469739416326902466659184, 7.896576152031804732350559159583, 8.526341491186007356861932140910, 9.585608222422790701792064957093, 11.04476330351685157171844998532, 11.48957226474368456594356960873

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