Properties

Label 2-384-128.101-c1-0-13
Degree $2$
Conductor $384$
Sign $0.888 + 0.458i$
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  + (−1.40 + 0.151i)2-s + (−0.881 + 0.471i)3-s + (1.95 − 0.425i)4-s + (0.410 + 0.500i)5-s + (1.16 − 0.796i)6-s + (3.77 − 2.52i)7-s + (−2.68 + 0.894i)8-s + (0.555 − 0.831i)9-s + (−0.653 − 0.641i)10-s + (−0.820 + 0.248i)11-s + (−1.52 + 1.29i)12-s + (−4.17 − 3.42i)13-s + (−4.92 + 4.11i)14-s + (−0.597 − 0.247i)15-s + (3.63 − 1.66i)16-s + (1.11 − 0.460i)17-s + ⋯
L(s)  = 1  + (−0.994 + 0.107i)2-s + (−0.509 + 0.272i)3-s + (0.977 − 0.212i)4-s + (0.183 + 0.223i)5-s + (0.477 − 0.325i)6-s + (1.42 − 0.953i)7-s + (−0.948 + 0.316i)8-s + (0.185 − 0.277i)9-s + (−0.206 − 0.202i)10-s + (−0.247 + 0.0750i)11-s + (−0.439 + 0.374i)12-s + (−1.15 − 0.950i)13-s + (−1.31 + 1.10i)14-s + (−0.154 − 0.0639i)15-s + (0.909 − 0.416i)16-s + (0.269 − 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821155 - 0.199217i\)
\(L(\frac12)\) \(\approx\) \(0.821155 - 0.199217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.151i)T \)
3 \( 1 + (0.881 - 0.471i)T \)
good5 \( 1 + (-0.410 - 0.500i)T + (-0.975 + 4.90i)T^{2} \)
7 \( 1 + (-3.77 + 2.52i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.820 - 0.248i)T + (9.14 - 6.11i)T^{2} \)
13 \( 1 + (4.17 + 3.42i)T + (2.53 + 12.7i)T^{2} \)
17 \( 1 + (-1.11 + 0.460i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.204 - 2.07i)T + (-18.6 + 3.70i)T^{2} \)
23 \( 1 + (-5.13 + 1.02i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-2.48 + 8.18i)T + (-24.1 - 16.1i)T^{2} \)
31 \( 1 + (-2.53 - 2.53i)T + 31iT^{2} \)
37 \( 1 + (-1.74 - 0.171i)T + (36.2 + 7.21i)T^{2} \)
41 \( 1 + (-1.07 - 5.38i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-7.39 - 3.95i)T + (23.8 + 35.7i)T^{2} \)
47 \( 1 + (0.0457 + 0.110i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (3.07 + 10.1i)T + (-44.0 + 29.4i)T^{2} \)
59 \( 1 + (0.373 - 0.306i)T + (11.5 - 57.8i)T^{2} \)
61 \( 1 + (-3.11 - 5.82i)T + (-33.8 + 50.7i)T^{2} \)
67 \( 1 + (1.96 + 3.68i)T + (-37.2 + 55.7i)T^{2} \)
71 \( 1 + (-5.37 - 8.04i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-0.580 - 0.388i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (4.16 - 10.0i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.25 - 0.419i)T + (81.4 - 16.1i)T^{2} \)
89 \( 1 + (13.0 + 2.60i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (0.365 + 0.365i)T + 97iT^{2} \)
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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

−11.03941400296491232627947857478, −10.28050630517775260665998745960, −9.784063136150540024920098950875, −8.256438078272190915866627196305, −7.72327937802742397338569214821, −6.77049914047200253129689479895, −5.49556034791958812769223484593, −4.51899074575823404429627830449, −2.62374514636572959932544096028, −0.928891697472640064467871428678, 1.46140613143240991382204806653, 2.57695626895882450818842231256, 4.81595571863617407341238206498, 5.61672548981951074642616834636, 6.98117933943789707008092571419, 7.69720632017903027377844154806, 8.813092186018345691336823769340, 9.340687249084151915112292828780, 10.67674207449575876251392119326, 11.30890946074104881911091530560

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