Properties

Label 2-384-48.29-c2-0-8
Degree $2$
Conductor $384$
Sign $0.600 - 0.799i$
Analytic cond. $10.4632$
Root an. cond. $3.23469$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.99 + 0.164i)3-s + (−3.61 − 3.61i)5-s + 12.2i·7-s + (8.94 + 0.985i)9-s + (1.76 + 1.76i)11-s + (2.38 + 2.38i)13-s + (−10.2 − 11.4i)15-s + 20.0i·17-s + (8.77 + 8.77i)19-s + (−2.02 + 36.7i)21-s + 13.1·23-s + 1.10i·25-s + (26.6 + 4.42i)27-s + (−6.51 + 6.51i)29-s + 37.5·31-s + ⋯
L(s)  = 1  + (0.998 + 0.0548i)3-s + (−0.722 − 0.722i)5-s + 1.75i·7-s + (0.993 + 0.109i)9-s + (0.160 + 0.160i)11-s + (0.183 + 0.183i)13-s + (−0.681 − 0.761i)15-s + 1.18i·17-s + (0.461 + 0.461i)19-s + (−0.0962 + 1.75i)21-s + 0.573·23-s + 0.0443i·25-s + (0.986 + 0.163i)27-s + (−0.224 + 0.224i)29-s + 1.21·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(10.4632\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1),\ 0.600 - 0.799i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88916 + 0.943535i\)
\(L(\frac12)\) \(\approx\) \(1.88916 + 0.943535i\)
\(L(2)\) 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 \( 1 + (-2.99 - 0.164i)T \)
good5 \( 1 + (3.61 + 3.61i)T + 25iT^{2} \)
7 \( 1 - 12.2iT - 49T^{2} \)
11 \( 1 + (-1.76 - 1.76i)T + 121iT^{2} \)
13 \( 1 + (-2.38 - 2.38i)T + 169iT^{2} \)
17 \( 1 - 20.0iT - 289T^{2} \)
19 \( 1 + (-8.77 - 8.77i)T + 361iT^{2} \)
23 \( 1 - 13.1T + 529T^{2} \)
29 \( 1 + (6.51 - 6.51i)T - 841iT^{2} \)
31 \( 1 - 37.5T + 961T^{2} \)
37 \( 1 + (10.0 - 10.0i)T - 1.36e3iT^{2} \)
41 \( 1 - 4.57T + 1.68e3T^{2} \)
43 \( 1 + (21.2 - 21.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 54.8iT - 2.20e3T^{2} \)
53 \( 1 + (-21.5 - 21.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (53.6 + 53.6i)T + 3.48e3iT^{2} \)
61 \( 1 + (-19.2 - 19.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (31.5 + 31.5i)T + 4.48e3iT^{2} \)
71 \( 1 - 65.1T + 5.04e3T^{2} \)
73 \( 1 - 50.2iT - 5.32e3T^{2} \)
79 \( 1 - 20.9T + 6.24e3T^{2} \)
83 \( 1 + (6.35 - 6.35i)T - 6.88e3iT^{2} \)
89 \( 1 + 166.T + 7.92e3T^{2} \)
97 \( 1 - 139.T + 9.40e3T^{2} \)
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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

−11.49454050308068839196205151765, −10.09487866785733417648113770765, −9.132630379034599988307117179440, −8.493528526692682759879603130827, −7.983912052646465561711771540751, −6.53805342535160153993757132491, −5.29366628758689054024631615938, −4.18113575914085733098177179253, −2.99382382499328499270667705706, −1.68366108747882099320568970592, 0.915345119695454790440431874652, 2.91982561975315305839089490374, 3.72324168828394867337606448788, 4.69318319876303503625278640601, 6.74083198280941000705162876178, 7.32263505765960904228868700016, 7.935225107236442233732740632416, 9.181188640910610021602947524853, 10.11216877238414957550410616846, 10.89276277879173885414742466626

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