Properties

Label 2-384-384.107-c1-0-50
Degree $2$
Conductor $384$
Sign $0.935 + 0.353i$
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  + (−0.833 + 1.14i)2-s + (1.36 − 1.06i)3-s + (−0.610 − 1.90i)4-s + (2.81 − 0.855i)5-s + (0.0806 + 2.44i)6-s + (0.140 − 0.707i)7-s + (2.68 + 0.888i)8-s + (0.726 − 2.91i)9-s + (−1.37 + 3.93i)10-s + (0.0513 − 0.521i)11-s + (−2.86 − 1.94i)12-s + (−3.43 − 1.04i)13-s + (0.691 + 0.750i)14-s + (2.93 − 4.17i)15-s + (−3.25 + 2.32i)16-s + (−5.36 + 2.22i)17-s + ⋯
L(s)  = 1  + (−0.589 + 0.807i)2-s + (0.788 − 0.615i)3-s + (−0.305 − 0.952i)4-s + (1.26 − 0.382i)5-s + (0.0329 + 0.999i)6-s + (0.0532 − 0.267i)7-s + (0.949 + 0.314i)8-s + (0.242 − 0.970i)9-s + (−0.433 + 1.24i)10-s + (0.0154 − 0.157i)11-s + (−0.826 − 0.562i)12-s + (−0.951 − 0.288i)13-s + (0.184 + 0.200i)14-s + (0.758 − 1.07i)15-s + (−0.813 + 0.581i)16-s + (−1.30 + 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.935 + 0.353i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.935 + 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47682 - 0.269534i\)
\(L(\frac12)\) \(\approx\) \(1.47682 - 0.269534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.833 - 1.14i)T \)
3 \( 1 + (-1.36 + 1.06i)T \)
good5 \( 1 + (-2.81 + 0.855i)T + (4.15 - 2.77i)T^{2} \)
7 \( 1 + (-0.140 + 0.707i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.0513 + 0.521i)T + (-10.7 - 2.14i)T^{2} \)
13 \( 1 + (3.43 + 1.04i)T + (10.8 + 7.22i)T^{2} \)
17 \( 1 + (5.36 - 2.22i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-5.34 + 2.85i)T + (10.5 - 15.7i)T^{2} \)
23 \( 1 + (-0.0781 + 0.116i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.731 - 7.42i)T + (-28.4 + 5.65i)T^{2} \)
31 \( 1 + (-1.32 + 1.32i)T - 31iT^{2} \)
37 \( 1 + (-5.74 - 3.07i)T + (20.5 + 30.7i)T^{2} \)
41 \( 1 + (0.746 - 1.11i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-3.03 - 3.70i)T + (-8.38 + 42.1i)T^{2} \)
47 \( 1 + (2.21 - 0.917i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (0.658 - 6.68i)T + (-51.9 - 10.3i)T^{2} \)
59 \( 1 + (-11.6 + 3.54i)T + (49.0 - 32.7i)T^{2} \)
61 \( 1 + (7.37 - 8.99i)T + (-11.9 - 59.8i)T^{2} \)
67 \( 1 + (8.46 + 6.94i)T + (13.0 + 65.7i)T^{2} \)
71 \( 1 + (0.480 - 2.41i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-11.5 + 2.30i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-0.628 + 1.51i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.238 - 0.127i)T + (46.1 - 69.0i)T^{2} \)
89 \( 1 + (10.2 - 6.88i)T + (34.0 - 82.2i)T^{2} \)
97 \( 1 + (12.0 + 12.0i)T + 97iT^{2} \)
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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

−11.01541349298409213542962384672, −9.879399044731671606625537152971, −9.326420330283586110281815349497, −8.579885062437547559057748785959, −7.51703171907412428539860552013, −6.73058457836732040678554879253, −5.76238369122127594010330128581, −4.63573264059714783292587837241, −2.56864153349987229402807836481, −1.26696727656408212360433102076, 2.07979239573304133702042921250, 2.70019597369814686833150031638, 4.16597267896438673578147018632, 5.33294011220355229928873075355, 6.92385568820933651403493501998, 7.951702752784779725644880187156, 9.077904794179076466553497944511, 9.661677472700225415009890349365, 10.12873273852490204270003462289, 11.16289956126431723310476483341

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