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Properties

Label 2-384-32.13-c1-0-0
Degree $2$
Conductor $384$
Sign $-0.964 + 0.264i$
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  + (−0.382 + 0.923i)3-s + (−1.20 + 0.498i)5-s + (−2.59 + 2.59i)7-s + (−0.707 − 0.707i)9-s + (−2.14 − 5.18i)11-s + (−0.984 − 0.407i)13-s − 1.30i·15-s − 0.979i·17-s + (−5.68 − 2.35i)19-s + (−1.40 − 3.38i)21-s + (3.70 + 3.70i)23-s + (−2.33 + 2.33i)25-s + (0.923 − 0.382i)27-s + (−1.17 + 2.83i)29-s − 1.54·31-s + ⋯
L(s)  = 1  + (−0.220 + 0.533i)3-s + (−0.538 + 0.223i)5-s + (−0.980 + 0.980i)7-s + (−0.235 − 0.235i)9-s + (−0.647 − 1.56i)11-s + (−0.272 − 0.113i)13-s − 0.336i·15-s − 0.237i·17-s + (−1.30 − 0.540i)19-s + (−0.306 − 0.739i)21-s + (0.771 + 0.771i)23-s + (−0.466 + 0.466i)25-s + (0.177 − 0.0736i)27-s + (−0.217 + 0.525i)29-s − 0.277·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.964 + 0.264i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.964 + 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0209797 - 0.156003i\)
\(L(\frac12)\) \(\approx\) \(0.0209797 - 0.156003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.382 - 0.923i)T \)
good5 \( 1 + (1.20 - 0.498i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.59 - 2.59i)T - 7iT^{2} \)
11 \( 1 + (2.14 + 5.18i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.984 + 0.407i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.979iT - 17T^{2} \)
19 \( 1 + (5.68 + 2.35i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.70 - 3.70i)T + 23iT^{2} \)
29 \( 1 + (1.17 - 2.83i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.54T + 31T^{2} \)
37 \( 1 + (8.23 - 3.41i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.10 + 1.10i)T + 41iT^{2} \)
43 \( 1 + (-3.47 - 8.37i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.15iT - 47T^{2} \)
53 \( 1 + (2.55 + 6.16i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-8.95 + 3.70i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.00 - 4.84i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.14 - 2.76i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (10.0 - 10.0i)T - 71iT^{2} \)
73 \( 1 + (-8.11 - 8.11i)T + 73iT^{2} \)
79 \( 1 + 0.155iT - 79T^{2} \)
83 \( 1 + (5.13 + 2.12i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (6.15 - 6.15i)T - 89iT^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−11.57708330253996297377696186926, −11.05468530640303737698551173449, −10.04797256739677500961306765307, −9.017865613095621671698470465168, −8.400048868951688501098692509094, −7.02981096712062110834864138118, −5.96310418729480907804596812947, −5.17691074614872562559863652090, −3.60356458784476948620552542171, −2.80912261364489766209276603629, 0.10005479971583221103833475987, 2.17140522393820653487157684458, 3.84002537136633791935847395793, 4.77095067276810375043706542082, 6.27288787160138828825418623904, 7.13892836043552859920114006275, 7.76376258043033134649087714716, 8.984817611738080465557923456830, 10.21776950246158150413682202286, 10.59826849695668615746208104610

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