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Properties

Label 2-384-128.93-c1-0-16
Degree $2$
Conductor $384$
Sign $-0.0750 - 0.997i$
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  + (0.946 + 1.05i)2-s + (0.995 + 0.0980i)3-s + (−0.209 + 1.98i)4-s + (0.250 + 0.467i)5-s + (0.838 + 1.13i)6-s + (0.134 + 0.676i)7-s + (−2.28 + 1.66i)8-s + (0.980 + 0.195i)9-s + (−0.255 + 0.705i)10-s + (0.842 + 0.691i)11-s + (−0.403 + 1.95i)12-s + (−0.235 − 0.125i)13-s + (−0.583 + 0.781i)14-s + (0.203 + 0.490i)15-s + (−3.91 − 0.832i)16-s + (0.140 − 0.339i)17-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (0.574 + 0.0565i)3-s + (−0.104 + 0.994i)4-s + (0.111 + 0.209i)5-s + (0.342 + 0.464i)6-s + (0.0508 + 0.255i)7-s + (−0.809 + 0.587i)8-s + (0.326 + 0.0650i)9-s + (−0.0806 + 0.223i)10-s + (0.254 + 0.208i)11-s + (−0.116 + 0.565i)12-s + (−0.0652 − 0.0349i)13-s + (−0.156 + 0.208i)14-s + (0.0524 + 0.126i)15-s + (−0.978 − 0.208i)16-s + (0.0340 − 0.0823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0750 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0750 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.0750 - 0.997i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.0750 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52815 + 1.64743i\)
\(L(\frac12)\) \(\approx\) \(1.52815 + 1.64743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.946 - 1.05i)T \)
3 \( 1 + (-0.995 - 0.0980i)T \)
good5 \( 1 + (-0.250 - 0.467i)T + (-2.77 + 4.15i)T^{2} \)
7 \( 1 + (-0.134 - 0.676i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.842 - 0.691i)T + (2.14 + 10.7i)T^{2} \)
13 \( 1 + (0.235 + 0.125i)T + (7.22 + 10.8i)T^{2} \)
17 \( 1 + (-0.140 + 0.339i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.45 + 0.442i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (1.18 - 0.790i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (0.269 + 0.328i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (0.349 - 0.349i)T - 31iT^{2} \)
37 \( 1 + (-1.19 + 3.94i)T + (-30.7 - 20.5i)T^{2} \)
41 \( 1 + (3.34 + 5.00i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-3.57 + 0.352i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (0.509 + 0.210i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (3.73 - 4.55i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (0.179 - 0.0957i)T + (32.7 - 49.0i)T^{2} \)
61 \( 1 + (-0.753 + 7.64i)T + (-59.8 - 11.9i)T^{2} \)
67 \( 1 + (0.177 - 1.79i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (10.7 - 2.13i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-2.58 + 12.9i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-0.435 + 0.180i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-3.49 - 11.5i)T + (-69.0 + 46.1i)T^{2} \)
89 \( 1 + (2.58 + 1.72i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-6.33 + 6.33i)T - 97iT^{2} \)
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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

−11.89279496175015467963038254675, −10.69604228772146227309642897042, −9.477530504130991453747352291353, −8.677763941059977238502188637229, −7.71815898743028408840850442397, −6.87316718324826163094663953733, −5.82845300546730481315437633771, −4.70359404212624467495492182397, −3.60071539911300743355017958766, −2.40721883172824125380718795111, 1.37694103898508502024932197251, 2.83339848856375802522708480397, 3.90752394770746315377276553382, 4.97375227810477758567364493712, 6.13225501073072406716446106303, 7.26958060226695354276750160571, 8.552080245929386863021014022983, 9.433814648883595190078057662434, 10.24123795991269203027550736159, 11.19995671331647634035647927867

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