Properties

Label 2-384-128.53-c1-0-24
Degree $2$
Conductor $384$
Sign $-0.948 + 0.318i$
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  + (−0.310 − 1.37i)2-s + (0.0980 + 0.995i)3-s + (−1.80 + 0.857i)4-s + (−1.10 − 0.592i)5-s + (1.34 − 0.444i)6-s + (0.457 − 2.29i)7-s + (1.74 + 2.22i)8-s + (−0.980 + 0.195i)9-s + (−0.472 + 1.71i)10-s + (−1.67 − 2.03i)11-s + (−1.03 − 1.71i)12-s + (−0.759 − 1.42i)13-s + (−3.31 + 0.0835i)14-s + (0.480 − 1.16i)15-s + (2.52 − 3.09i)16-s + (−1.47 − 3.56i)17-s + ⋯
L(s)  = 1  + (−0.219 − 0.975i)2-s + (0.0565 + 0.574i)3-s + (−0.903 + 0.428i)4-s + (−0.495 − 0.264i)5-s + (0.548 − 0.181i)6-s + (0.172 − 0.868i)7-s + (0.616 + 0.787i)8-s + (−0.326 + 0.0650i)9-s + (−0.149 + 0.541i)10-s + (−0.503 − 0.613i)11-s + (−0.297 − 0.494i)12-s + (−0.210 − 0.394i)13-s + (−0.885 + 0.0223i)14-s + (0.124 − 0.299i)15-s + (0.632 − 0.774i)16-s + (−0.358 − 0.865i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.948 + 0.318i$
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: \((2,\ 384,\ (\ :1/2),\ -0.948 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100220 - 0.613512i\)
\(L(\frac12)\) \(\approx\) \(0.100220 - 0.613512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.310 + 1.37i)T \)
3 \( 1 + (-0.0980 - 0.995i)T \)
good5 \( 1 + (1.10 + 0.592i)T + (2.77 + 4.15i)T^{2} \)
7 \( 1 + (-0.457 + 2.29i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (1.67 + 2.03i)T + (-2.14 + 10.7i)T^{2} \)
13 \( 1 + (0.759 + 1.42i)T + (-7.22 + 10.8i)T^{2} \)
17 \( 1 + (1.47 + 3.56i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (1.05 - 3.47i)T + (-15.7 - 10.5i)T^{2} \)
23 \( 1 + (5.89 + 3.93i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (5.69 + 4.67i)T + (5.65 + 28.4i)T^{2} \)
31 \( 1 + (0.0207 + 0.0207i)T + 31iT^{2} \)
37 \( 1 + (-9.61 + 2.91i)T + (30.7 - 20.5i)T^{2} \)
41 \( 1 + (-0.892 + 1.33i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (1.13 - 11.4i)T + (-42.1 - 8.38i)T^{2} \)
47 \( 1 + (-6.47 + 2.68i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-1.16 + 0.955i)T + (10.3 - 51.9i)T^{2} \)
59 \( 1 + (3.77 - 7.06i)T + (-32.7 - 49.0i)T^{2} \)
61 \( 1 + (-4.20 + 0.414i)T + (59.8 - 11.9i)T^{2} \)
67 \( 1 + (6.18 - 0.609i)T + (65.7 - 13.0i)T^{2} \)
71 \( 1 + (-8.04 - 1.60i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (2.06 + 10.3i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-6.65 - 2.75i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-9.78 - 2.96i)T + (69.0 + 46.1i)T^{2} \)
89 \( 1 + (6.96 - 4.65i)T + (34.0 - 82.2i)T^{2} \)
97 \( 1 + (0.488 + 0.488i)T + 97iT^{2} \)
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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

−10.87520666781418128329934768431, −10.16552242832277548238826656013, −9.356738426173191593895755095625, −8.151775090236225715393088403436, −7.71978612562262659733756752648, −5.82908188016170688132080074228, −4.49124242823389240178683640579, −3.90119134864571988124031298499, −2.53086079872894380062899081799, −0.43416890032124957999453827617, 2.05815419626123690327849309739, 3.92512618548289482892603430754, 5.22565548783143414615312825200, 6.14281903554730605316158584774, 7.20541769900713729745188669161, 7.84706038046684165169235000980, 8.787109817730866104312864110363, 9.601840811166268349553836839007, 10.82340156114235477588993726856, 11.84228621769435821062054221350

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