Properties

Label 2-384-128.29-c1-0-5
Degree $2$
Conductor $384$
Sign $-0.951 - 0.308i$
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  + (0.886 + 1.10i)2-s + (0.0980 − 0.995i)3-s + (−0.428 + 1.95i)4-s + (−3.03 + 1.62i)5-s + (1.18 − 0.774i)6-s + (−0.0609 − 0.306i)7-s + (−2.53 + 1.25i)8-s + (−0.980 − 0.195i)9-s + (−4.48 − 1.90i)10-s + (−3.52 + 4.30i)11-s + (1.90 + 0.617i)12-s + (0.800 − 1.49i)13-s + (0.283 − 0.338i)14-s + (1.31 + 3.18i)15-s + (−3.63 − 1.67i)16-s + (−0.777 + 1.87i)17-s + ⋯
L(s)  = 1  + (0.626 + 0.779i)2-s + (0.0565 − 0.574i)3-s + (−0.214 + 0.976i)4-s + (−1.35 + 0.726i)5-s + (0.483 − 0.316i)6-s + (−0.0230 − 0.115i)7-s + (−0.895 + 0.445i)8-s + (−0.326 − 0.0650i)9-s + (−1.41 − 0.603i)10-s + (−1.06 + 1.29i)11-s + (0.549 + 0.178i)12-s + (0.222 − 0.415i)13-s + (0.0758 − 0.0905i)14-s + (0.340 + 0.821i)15-s + (−0.908 − 0.418i)16-s + (−0.188 + 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.951 - 0.308i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.951 - 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148390 + 0.938674i\)
\(L(\frac12)\) \(\approx\) \(0.148390 + 0.938674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.886 - 1.10i)T \)
3 \( 1 + (-0.0980 + 0.995i)T \)
good5 \( 1 + (3.03 - 1.62i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (0.0609 + 0.306i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (3.52 - 4.30i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (-0.800 + 1.49i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (0.777 - 1.87i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.40 - 4.63i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (-2.04 + 1.36i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-2.41 + 1.98i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (-1.08 + 1.08i)T - 31iT^{2} \)
37 \( 1 + (5.25 + 1.59i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (-5.31 - 7.95i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-1.10 - 11.2i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (7.30 + 3.02i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-1.14 - 0.942i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (0.278 + 0.520i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (-14.9 - 1.47i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (10.7 + 1.05i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (-2.97 + 0.591i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.34 + 6.76i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (10.0 - 4.16i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (3.38 - 1.02i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (-9.10 - 6.08i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (1.18 - 1.18i)T - 97iT^{2} \)
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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

−12.01030283811962775259304665356, −11.09881605019861055993007390842, −10.00477570600101691543826977252, −8.357545524133297206676779615125, −7.77256362882546385773519014195, −7.19615674404493240393960175364, −6.21753093613182684641587188628, −4.88493004456848483689164927178, −3.81141038725759640972357291797, −2.69915372545726363303873807709, 0.50479550414844983764845176249, 2.86753813622426657021395304579, 3.80519364907262196221506775225, 4.82502233640834620832703605610, 5.55212458557834042517813221203, 7.16316855848453494954572091375, 8.568353546593965369911752700266, 8.962345850129865447181505619487, 10.33946263603674907980398879672, 11.18413633629991643263712639206

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