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Properties

Label 2-384-16.5-c1-0-6
Degree $2$
Conductor $384$
Sign $-0.0734 + 0.997i$
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  + (0.707 − 0.707i)3-s + (−1.27 − 1.27i)5-s + 0.158i·7-s − 1.00i·9-s + (−3.79 − 3.79i)11-s + (4.21 − 4.21i)13-s − 1.79·15-s + 3.05·17-s + (−2.15 + 2.15i)19-s + (0.112 + 0.112i)21-s − 2.82i·23-s − 1.76i·25-s + (−0.707 − 0.707i)27-s + (−2.09 + 2.09i)29-s − 4.15·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.568 − 0.568i)5-s + 0.0600i·7-s − 0.333i·9-s + (−1.14 − 1.14i)11-s + (1.16 − 1.16i)13-s − 0.464·15-s + 0.740·17-s + (−0.495 + 0.495i)19-s + (0.0245 + 0.0245i)21-s − 0.589i·23-s − 0.353i·25-s + (−0.136 − 0.136i)27-s + (−0.389 + 0.389i)29-s − 0.746·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0734 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.0734 + 0.997i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.0734 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862864 - 0.928724i\)
\(L(\frac12)\) \(\approx\) \(0.862864 - 0.928724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (1.27 + 1.27i)T + 5iT^{2} \)
7 \( 1 - 0.158iT - 7T^{2} \)
11 \( 1 + (3.79 + 3.79i)T + 11iT^{2} \)
13 \( 1 + (-4.21 + 4.21i)T - 13iT^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 + (2.15 - 2.15i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (2.09 - 2.09i)T - 29iT^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 + (-5.98 - 5.98i)T + 37iT^{2} \)
41 \( 1 + 2.60iT - 41T^{2} \)
43 \( 1 + (-5.75 - 5.75i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (3.55 + 3.55i)T + 53iT^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 + (3.66 - 3.66i)T - 61iT^{2} \)
67 \( 1 + (-0.767 + 0.767i)T - 67iT^{2} \)
71 \( 1 + 0.317iT - 71T^{2} \)
73 \( 1 + 1.33iT - 73T^{2} \)
79 \( 1 - 9.69T + 79T^{2} \)
83 \( 1 + (-0.115 + 0.115i)T - 83iT^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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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

−11.00491533468650374821257111830, −10.37880314905146729576214645722, −8.950883994620629193637390458467, −8.137892846510778521745843040446, −7.81753599269061362278311031016, −6.18868490680495513156890602843, −5.37910099614900673517665001160, −3.86161523887856243358622988348, −2.83021223977478145446916601254, −0.837226976059706259047394949791, 2.16088660777696605820702364378, 3.55113341355270170593859663200, 4.45080005763411113761176764510, 5.75886174327497557035177111877, 7.15430293212040656046739945648, 7.71635128749641900934273557370, 8.907635880876023686887265984768, 9.732496570066500375526632267243, 10.77700748297874100083368081954, 11.30332697991790147512699806151

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