Properties

Label 2-384-384.275-c1-0-6
Degree $2$
Conductor $384$
Sign $-0.229 - 0.973i$
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  + (−0.424 − 1.34i)2-s + (−0.796 + 1.53i)3-s + (−1.63 + 1.14i)4-s + (2.61 + 2.14i)5-s + (2.41 + 0.421i)6-s + (−1.02 + 1.53i)7-s + (2.24 + 1.72i)8-s + (−1.73 − 2.44i)9-s + (1.78 − 4.44i)10-s + (−4.07 − 1.23i)11-s + (−0.455 − 3.43i)12-s + (−0.147 + 0.121i)13-s + (2.51 + 0.734i)14-s + (−5.39 + 2.31i)15-s + (1.37 − 3.75i)16-s + (−2.33 + 5.63i)17-s + ⋯
L(s)  = 1  + (−0.300 − 0.953i)2-s + (−0.459 + 0.888i)3-s + (−0.819 + 0.572i)4-s + (1.17 + 0.961i)5-s + (0.985 + 0.172i)6-s + (−0.388 + 0.581i)7-s + (0.792 + 0.610i)8-s + (−0.577 − 0.816i)9-s + (0.565 − 1.40i)10-s + (−1.22 − 0.372i)11-s + (−0.131 − 0.991i)12-s + (−0.0410 + 0.0336i)13-s + (0.671 + 0.196i)14-s + (−1.39 + 0.598i)15-s + (0.344 − 0.938i)16-s + (−0.566 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440496 + 0.556698i\)
\(L(\frac12)\) \(\approx\) \(0.440496 + 0.556698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.424 + 1.34i)T \)
3 \( 1 + (0.796 - 1.53i)T \)
good5 \( 1 + (-2.61 - 2.14i)T + (0.975 + 4.90i)T^{2} \)
7 \( 1 + (1.02 - 1.53i)T + (-2.67 - 6.46i)T^{2} \)
11 \( 1 + (4.07 + 1.23i)T + (9.14 + 6.11i)T^{2} \)
13 \( 1 + (0.147 - 0.121i)T + (2.53 - 12.7i)T^{2} \)
17 \( 1 + (2.33 - 5.63i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (7.96 + 0.784i)T + (18.6 + 3.70i)T^{2} \)
23 \( 1 + (-4.90 - 0.974i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-6.87 + 2.08i)T + (24.1 - 16.1i)T^{2} \)
31 \( 1 + (-1.59 - 1.59i)T + 31iT^{2} \)
37 \( 1 + (11.3 - 1.11i)T + (36.2 - 7.21i)T^{2} \)
41 \( 1 + (-3.55 - 0.707i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.907 - 1.69i)T + (-23.8 + 35.7i)T^{2} \)
47 \( 1 + (2.29 - 5.53i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (-0.969 - 0.294i)T + (44.0 + 29.4i)T^{2} \)
59 \( 1 + (-4.29 - 3.52i)T + (11.5 + 57.8i)T^{2} \)
61 \( 1 + (3.08 - 5.76i)T + (-33.8 - 50.7i)T^{2} \)
67 \( 1 + (-2.22 - 1.18i)T + (37.2 + 55.7i)T^{2} \)
71 \( 1 + (-1.94 + 2.90i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-7.78 + 5.19i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (7.29 - 3.02i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-10.1 - 0.995i)T + (81.4 + 16.1i)T^{2} \)
89 \( 1 + (0.824 + 4.14i)T + (-82.2 + 34.0i)T^{2} \)
97 \( 1 + (-6.89 + 6.89i)T - 97iT^{2} \)
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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

−11.12591207157708834368893838703, −10.47114222099369674360284398318, −10.27760708237411762586172485233, −9.096165979071506302144247821060, −8.421317998318242541115117452814, −6.56904471796140078000647908508, −5.75342708537608933161558562611, −4.59978163467101299515465258799, −3.15264745149238247804719541910, −2.30587656074065091168215981302, 0.52692626932915373907899239153, 2.17121829101047276548733324125, 4.80838421770373263300866030793, 5.29675945119117204654550746036, 6.49000524887478857244908701601, 7.06750192436793900939388649301, 8.276567325472443093982904258994, 8.985490475759418794761949287660, 10.12430750311813547065474275223, 10.72925854164215417794937844204

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