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Properties

Label 2-384-48.35-c3-0-30
Degree $2$
Conductor $384$
Sign $-0.300 + 0.953i$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.86 + 3.47i)3-s + (13.5 + 13.5i)5-s − 19.7·7-s + (2.80 − 26.8i)9-s + (−20.5 + 20.5i)11-s + (−36.7 − 36.7i)13-s + (−99.6 − 5.19i)15-s + 4.20i·17-s + (−38.6 + 38.6i)19-s + (76.1 − 68.6i)21-s − 69.4i·23-s + 243. i·25-s + (82.5 + 113. i)27-s + (23.0 − 23.0i)29-s − 219. i·31-s + ⋯
L(s)  = 1  + (−0.742 + 0.669i)3-s + (1.21 + 1.21i)5-s − 1.06·7-s + (0.103 − 0.994i)9-s + (−0.562 + 0.562i)11-s + (−0.783 − 0.783i)13-s + (−1.71 − 0.0893i)15-s + 0.0599i·17-s + (−0.466 + 0.466i)19-s + (0.791 − 0.713i)21-s − 0.629i·23-s + 1.95i·25-s + (0.588 + 0.808i)27-s + (0.147 − 0.147i)29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.300 + 0.953i$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ -0.300 + 0.953i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.04353570028\)
\(L(\frac12)\) \(\approx\) \(0.04353570028\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.86 - 3.47i)T \)
good5 \( 1 + (-13.5 - 13.5i)T + 125iT^{2} \)
7 \( 1 + 19.7T + 343T^{2} \)
11 \( 1 + (20.5 - 20.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (36.7 + 36.7i)T + 2.19e3iT^{2} \)
17 \( 1 - 4.20iT - 4.91e3T^{2} \)
19 \( 1 + (38.6 - 38.6i)T - 6.85e3iT^{2} \)
23 \( 1 + 69.4iT - 1.21e4T^{2} \)
29 \( 1 + (-23.0 + 23.0i)T - 2.43e4iT^{2} \)
31 \( 1 + 219. iT - 2.97e4T^{2} \)
37 \( 1 + (68.0 - 68.0i)T - 5.06e4iT^{2} \)
41 \( 1 - 325.T + 6.89e4T^{2} \)
43 \( 1 + (-36.9 - 36.9i)T + 7.95e4iT^{2} \)
47 \( 1 + 192.T + 1.03e5T^{2} \)
53 \( 1 + (461. + 461. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-0.977 + 0.977i)T - 2.05e5iT^{2} \)
61 \( 1 + (216. + 216. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-27.8 + 27.8i)T - 3.00e5iT^{2} \)
71 \( 1 + 786. iT - 3.57e5T^{2} \)
73 \( 1 + 510. iT - 3.89e5T^{2} \)
79 \( 1 - 230. iT - 4.93e5T^{2} \)
83 \( 1 + (593. + 593. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 805.T + 9.12e5T^{2} \)
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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

−10.30260080600657524444876830976, −10.05210129438950056069206544903, −9.337503463363751566515839061943, −7.58157938062867876722253801199, −6.44048530756919868266995657757, −6.03020152820346655094476332386, −4.88290454317128340940553406682, −3.35900104878093009681123854950, −2.36141007542203322392243937708, −0.01595093299925941553978829318, 1.34507603497285595474348118013, 2.60715342069503570792246213753, 4.64034148322309248297845423172, 5.52069438972251752166821120298, 6.26731965943315358651224250159, 7.21158363242555323777033701655, 8.565971902498197898012632230909, 9.410751257012329255970355569058, 10.17137426400639497722856784007, 11.24063762266821382214465699920

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