Properties

Label 2-384-48.35-c1-0-11
Degree $2$
Conductor $384$
Sign $-0.855 + 0.517i$
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  + (0.814 − 1.52i)3-s + (−2.08 − 2.08i)5-s − 1.14·7-s + (−1.67 − 2.48i)9-s + (−1.67 + 1.67i)11-s + (−0.146 − 0.146i)13-s + (−4.88 + 1.48i)15-s − 5.59i·17-s + (−1.48 + 1.48i)19-s + (−0.933 + 1.75i)21-s + 3.34i·23-s + 3.68i·25-s + (−5.16 + 0.533i)27-s + (3.51 − 3.51i)29-s − 5.83i·31-s + ⋯
L(s)  = 1  + (0.470 − 0.882i)3-s + (−0.931 − 0.931i)5-s − 0.433·7-s + (−0.558 − 0.829i)9-s + (−0.504 + 0.504i)11-s + (−0.0405 − 0.0405i)13-s + (−1.26 + 0.384i)15-s − 1.35i·17-s + (−0.341 + 0.341i)19-s + (−0.203 + 0.382i)21-s + 0.698i·23-s + 0.737i·25-s + (−0.994 + 0.102i)27-s + (0.652 − 0.652i)29-s − 1.04i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.855 + 0.517i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252245 - 0.905397i\)
\(L(\frac12)\) \(\approx\) \(0.252245 - 0.905397i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.814 + 1.52i)T \)
good5 \( 1 + (2.08 + 2.08i)T + 5iT^{2} \)
7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 + (1.67 - 1.67i)T - 11iT^{2} \)
13 \( 1 + (0.146 + 0.146i)T + 13iT^{2} \)
17 \( 1 + 5.59iT - 17T^{2} \)
19 \( 1 + (1.48 - 1.48i)T - 19iT^{2} \)
23 \( 1 - 3.34iT - 23T^{2} \)
29 \( 1 + (-3.51 + 3.51i)T - 29iT^{2} \)
31 \( 1 + 5.83iT - 31T^{2} \)
37 \( 1 + (-4.83 + 4.83i)T - 37iT^{2} \)
41 \( 1 - 0.610T + 41T^{2} \)
43 \( 1 + (-1.48 - 1.48i)T + 43iT^{2} \)
47 \( 1 - 6.41T + 47T^{2} \)
53 \( 1 + (0.164 + 0.164i)T + 53iT^{2} \)
59 \( 1 + (-9.05 + 9.05i)T - 59iT^{2} \)
61 \( 1 + (4.53 + 4.53i)T + 61iT^{2} \)
67 \( 1 + (-0.635 + 0.635i)T - 67iT^{2} \)
71 \( 1 + 6.90iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 - 9.83iT - 79T^{2} \)
83 \( 1 + (-8.09 - 8.09i)T + 83iT^{2} \)
89 \( 1 - 0.490T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−11.30649220125631692343772836345, −9.759478333270295814196608036105, −9.039197734371410348628101727690, −7.928917602065858895633503947760, −7.53863032695253235230963458187, −6.31401213712017284280230099877, −5.00219285093324934517258663671, −3.80261631814939999472667908199, −2.42684058434564036129609477098, −0.57960252942503700806024779729, 2.73759855601906288781256491800, 3.55802835446090136128909585342, 4.56533818726268321568573755416, 5.99229308267491378061193707492, 7.10649584301642006276172147168, 8.194167844906370733476762757282, 8.814025516642146743704812018158, 10.21340285262470598333645343492, 10.62219800298338189055517349116, 11.42104432045100858428432898245

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