Properties

Label 2-384-384.323-c1-0-61
Degree $2$
Conductor $384$
Sign $-0.739 - 0.673i$
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  + (0.348 − 1.37i)2-s + (0.545 − 1.64i)3-s + (−1.75 − 0.955i)4-s + (−2.82 − 0.856i)5-s + (−2.06 − 1.32i)6-s + (−0.0745 − 0.374i)7-s + (−1.92 + 2.07i)8-s + (−2.40 − 1.79i)9-s + (−2.15 + 3.57i)10-s + (0.231 + 2.34i)11-s + (−2.52 + 2.36i)12-s + (3.12 − 0.949i)13-s + (−0.539 − 0.0285i)14-s + (−2.94 + 4.17i)15-s + (2.17 + 3.35i)16-s + (−0.745 − 0.308i)17-s + ⋯
L(s)  = 1  + (0.246 − 0.969i)2-s + (0.314 − 0.949i)3-s + (−0.878 − 0.477i)4-s + (−1.26 − 0.383i)5-s + (−0.842 − 0.539i)6-s + (−0.0281 − 0.141i)7-s + (−0.679 + 0.733i)8-s + (−0.801 − 0.597i)9-s + (−0.682 + 1.12i)10-s + (0.0696 + 0.707i)11-s + (−0.730 + 0.683i)12-s + (0.868 − 0.263i)13-s + (−0.144 − 0.00761i)14-s + (−0.761 + 1.07i)15-s + (0.543 + 0.839i)16-s + (−0.180 − 0.0748i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.739 - 0.673i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307494 + 0.794539i\)
\(L(\frac12)\) \(\approx\) \(0.307494 + 0.794539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.348 + 1.37i)T \)
3 \( 1 + (-0.545 + 1.64i)T \)
good5 \( 1 + (2.82 + 0.856i)T + (4.15 + 2.77i)T^{2} \)
7 \( 1 + (0.0745 + 0.374i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.231 - 2.34i)T + (-10.7 + 2.14i)T^{2} \)
13 \( 1 + (-3.12 + 0.949i)T + (10.8 - 7.22i)T^{2} \)
17 \( 1 + (0.745 + 0.308i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (-0.936 - 0.500i)T + (10.5 + 15.7i)T^{2} \)
23 \( 1 + (5.22 + 7.82i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (-0.407 + 4.14i)T + (-28.4 - 5.65i)T^{2} \)
31 \( 1 + (2.67 + 2.67i)T + 31iT^{2} \)
37 \( 1 + (5.34 - 2.85i)T + (20.5 - 30.7i)T^{2} \)
41 \( 1 + (1.74 + 2.61i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (-5.57 + 6.78i)T + (-8.38 - 42.1i)T^{2} \)
47 \( 1 + (10.9 + 4.55i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-0.956 - 9.71i)T + (-51.9 + 10.3i)T^{2} \)
59 \( 1 + (-11.2 - 3.40i)T + (49.0 + 32.7i)T^{2} \)
61 \( 1 + (-1.86 - 2.27i)T + (-11.9 + 59.8i)T^{2} \)
67 \( 1 + (-1.44 + 1.18i)T + (13.0 - 65.7i)T^{2} \)
71 \( 1 + (0.884 + 4.44i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-3.73 - 0.742i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (0.353 + 0.853i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (6.40 + 3.42i)T + (46.1 + 69.0i)T^{2} \)
89 \( 1 + (-5.43 - 3.63i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (8.45 - 8.45i)T - 97iT^{2} \)
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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

−11.05673389974507382731119838012, −10.01735890792179961310831106157, −8.688201018567975246152294980229, −8.253593360207085538057463185683, −7.14237291613049428896512526396, −5.86679036109772507653363247464, −4.39451083164968519161623454425, −3.56953860649744176145599534873, −2.12395962549744058313011746122, −0.51148537584534636442848842165, 3.43654025546543611698127474197, 3.78798332537938199701108804219, 5.06996273840154311226803996122, 6.11462438846017894837050147383, 7.36311578604107437213163757205, 8.217255402054815433480576262915, 8.854434272623988379017255899893, 9.881022437448266711636656520908, 11.16283867057744055263309074720, 11.64546657245668574483996452457

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