Properties

Label 2-384-384.131-c1-0-47
Degree $2$
Conductor $384$
Sign $0.961 + 0.273i$
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  + (1.38 − 0.297i)2-s + (1.72 − 0.0927i)3-s + (1.82 − 0.823i)4-s + (−0.606 + 1.99i)5-s + (2.36 − 0.643i)6-s + (−0.441 − 2.22i)7-s + (2.27 − 1.68i)8-s + (2.98 − 0.320i)9-s + (−0.242 + 2.94i)10-s + (−4.48 + 0.441i)11-s + (3.07 − 1.59i)12-s + (−0.494 − 1.63i)13-s + (−1.27 − 2.93i)14-s + (−0.863 + 3.51i)15-s + (2.64 − 3.00i)16-s + (2.01 + 0.835i)17-s + ⋯
L(s)  = 1  + (0.977 − 0.210i)2-s + (0.998 − 0.0535i)3-s + (0.911 − 0.411i)4-s + (−0.271 + 0.893i)5-s + (0.964 − 0.262i)6-s + (−0.167 − 0.839i)7-s + (0.804 − 0.594i)8-s + (0.994 − 0.106i)9-s + (−0.0767 + 0.930i)10-s + (−1.35 + 0.133i)11-s + (0.887 − 0.460i)12-s + (−0.137 − 0.452i)13-s + (−0.340 − 0.785i)14-s + (−0.222 + 0.907i)15-s + (0.660 − 0.750i)16-s + (0.489 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.961 + 0.273i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96798 - 0.413501i\)
\(L(\frac12)\) \(\approx\) \(2.96798 - 0.413501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.297i)T \)
3 \( 1 + (-1.72 + 0.0927i)T \)
good5 \( 1 + (0.606 - 1.99i)T + (-4.15 - 2.77i)T^{2} \)
7 \( 1 + (0.441 + 2.22i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (4.48 - 0.441i)T + (10.7 - 2.14i)T^{2} \)
13 \( 1 + (0.494 + 1.63i)T + (-10.8 + 7.22i)T^{2} \)
17 \( 1 + (-2.01 - 0.835i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (4.00 - 7.50i)T + (-10.5 - 15.7i)T^{2} \)
23 \( 1 + (-0.408 - 0.611i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (5.82 + 0.573i)T + (28.4 + 5.65i)T^{2} \)
31 \( 1 + (-0.361 - 0.361i)T + 31iT^{2} \)
37 \( 1 + (3.66 + 6.84i)T + (-20.5 + 30.7i)T^{2} \)
41 \( 1 + (-5.77 - 8.63i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (8.33 + 6.83i)T + (8.38 + 42.1i)T^{2} \)
47 \( 1 + (11.8 + 4.90i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.630 - 0.0621i)T + (51.9 - 10.3i)T^{2} \)
59 \( 1 + (-1.27 + 4.20i)T + (-49.0 - 32.7i)T^{2} \)
61 \( 1 + (-3.02 + 2.47i)T + (11.9 - 59.8i)T^{2} \)
67 \( 1 + (-3.91 - 4.77i)T + (-13.0 + 65.7i)T^{2} \)
71 \( 1 + (-1.06 - 5.35i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-6.32 - 1.25i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-4.60 - 11.1i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-3.39 + 6.34i)T + (-46.1 - 69.0i)T^{2} \)
89 \( 1 + (1.07 + 0.715i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (0.135 - 0.135i)T - 97iT^{2} \)
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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

−11.20115360858137284801946756598, −10.30016501798310819630408401334, −10.03464318318561122662685092867, −8.103823005584032179570518098685, −7.52782875247472240216300472549, −6.65192083268965837012825487978, −5.32188253985561232129298495162, −3.88709593730068881667340894384, −3.27946304592268986851499920362, −2.04863665328927108846379872379, 2.20534147686101562215152911986, 3.15136282813612435261002616443, 4.59035126967389748314058080467, 5.18604878999742098298062319959, 6.61515914767153212675434724657, 7.72479634035443259593589408542, 8.488511717780267059457953987511, 9.286833551572575692002032589220, 10.57998457149082758722692360712, 11.64947908558541995933605808297

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