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Properties

Label 2-384-16.13-c1-0-3
Degree $2$
Conductor $384$
Sign $0.997 - 0.0734i$
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

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.997 - 0.0734i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.997 - 0.0734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19901 + 0.0440762i\)
\(L(\frac12)\) \(\approx\) \(1.19901 + 0.0440762i\)
\(L(\frac{3}{2})\) 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

−11.52280469931592856654072360643, −10.76045541694562703483407656293, −9.453022332278949218479344460163, −8.517746801576729049849468914927, −7.54490120814462986085648514933, −6.45980578933270920062565077221, −5.91079996473555479231026452536, −4.20965764239114577503413450834, −3.25959903538321491543264066094, −1.29448788262382590148833226916, 1.13638174546973384585626250038, 3.39846258282795762731619665786, 4.37057186045183257531550490965, 5.38959363307017260177586280217, 6.51364601283883011684512838152, 7.65233355533570075360143246824, 8.601410138278508604608802376063, 9.578645898195727476419710260061, 10.37510129497643654720491448257, 11.48258966907837180170483891758

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