Properties

Label 2-384-32.29-c1-0-4
Degree $2$
Conductor $384$
Sign $0.756 - 0.654i$
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  + (0.923 + 0.382i)3-s + (1.36 + 3.28i)5-s + (2.73 − 2.73i)7-s + (0.707 + 0.707i)9-s + (−3.01 + 1.24i)11-s + (0.932 − 2.25i)13-s + 3.55i·15-s + 0.517i·17-s + (−1.52 + 3.68i)19-s + (3.56 − 1.47i)21-s + (−2.39 − 2.39i)23-s + (−5.42 + 5.42i)25-s + (0.382 + 0.923i)27-s + (7.09 + 2.93i)29-s + 1.50·31-s + ⋯
L(s)  = 1  + (0.533 + 0.220i)3-s + (0.609 + 1.47i)5-s + (1.03 − 1.03i)7-s + (0.235 + 0.235i)9-s + (−0.909 + 0.376i)11-s + (0.258 − 0.624i)13-s + 0.918i·15-s + 0.125i·17-s + (−0.350 + 0.845i)19-s + (0.778 − 0.322i)21-s + (−0.500 − 0.500i)23-s + (−1.08 + 1.08i)25-s + (0.0736 + 0.177i)27-s + (1.31 + 0.545i)29-s + 0.269·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.756 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74166 + 0.649043i\)
\(L(\frac12)\) \(\approx\) \(1.74166 + 0.649043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.923 - 0.382i)T \)
good5 \( 1 + (-1.36 - 3.28i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.73 + 2.73i)T - 7iT^{2} \)
11 \( 1 + (3.01 - 1.24i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.932 + 2.25i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.517iT - 17T^{2} \)
19 \( 1 + (1.52 - 3.68i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.39 + 2.39i)T + 23iT^{2} \)
29 \( 1 + (-7.09 - 2.93i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 + (3.40 + 8.22i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.21 + 3.21i)T + 41iT^{2} \)
43 \( 1 + (-1.31 + 0.544i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 4.67iT - 47T^{2} \)
53 \( 1 + (4.19 - 1.73i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.680 - 1.64i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-6.71 - 2.78i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (11.1 + 4.61i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (1.86 - 1.86i)T - 71iT^{2} \)
73 \( 1 + (9.06 + 9.06i)T + 73iT^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 + (-1.89 + 4.57i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.70 - 2.70i)T - 89iT^{2} \)
97 \( 1 - 3.73T + 97T^{2} \)
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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

−10.96984032603535934137852481099, −10.41794777182496557962661716709, −10.11349143284023620476088963216, −8.490969118273401127552952527464, −7.67619030270525864950223812691, −6.91857501944621620430808915502, −5.67471476548067809749716503287, −4.36623069554798264594632212055, −3.15491365846750079467017025573, −1.97781063820257721150887267200, 1.46888908848936502414235272948, 2.59935825225491293863564610543, 4.58012802966865126216262866338, 5.21515461143487860167822737104, 6.29782939018470635512161162913, 7.943203234179302628363395217735, 8.530340273379452909639542469908, 9.092868761973186474348041198184, 10.12396106957994164836087194932, 11.50823266013733161535657559816

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