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Properties

Label 2-384-128.37-c1-0-13
Degree $2$
Conductor $384$
Sign $-0.656 - 0.754i$
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  + (1.15 + 0.821i)2-s + (0.471 + 0.881i)3-s + (0.649 + 1.89i)4-s + (−1.38 + 1.13i)5-s + (−0.182 + 1.40i)6-s + (−1.43 + 0.960i)7-s + (−0.807 + 2.71i)8-s + (−0.555 + 0.831i)9-s + (−2.53 + 0.170i)10-s + (−0.677 − 2.23i)11-s + (−1.36 + 1.46i)12-s + (−0.399 + 0.487i)13-s + (−2.44 − 0.0758i)14-s + (−1.65 − 0.686i)15-s + (−3.15 + 2.45i)16-s + (2.54 − 1.05i)17-s + ⋯
L(s)  = 1  + (0.813 + 0.581i)2-s + (0.272 + 0.509i)3-s + (0.324 + 0.945i)4-s + (−0.620 + 0.509i)5-s + (−0.0743 + 0.572i)6-s + (−0.543 + 0.363i)7-s + (−0.285 + 0.958i)8-s + (−0.185 + 0.277i)9-s + (−0.800 + 0.0538i)10-s + (−0.204 − 0.673i)11-s + (−0.393 + 0.422i)12-s + (−0.110 + 0.135i)13-s + (−0.653 − 0.0202i)14-s + (−0.428 − 0.177i)15-s + (−0.789 + 0.614i)16-s + (0.618 − 0.256i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.656 - 0.754i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.656 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.791956 + 1.73788i\)
\(L(\frac12)\) \(\approx\) \(0.791956 + 1.73788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.821i)T \)
3 \( 1 + (-0.471 - 0.881i)T \)
good5 \( 1 + (1.38 - 1.13i)T + (0.975 - 4.90i)T^{2} \)
7 \( 1 + (1.43 - 0.960i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.677 + 2.23i)T + (-9.14 + 6.11i)T^{2} \)
13 \( 1 + (0.399 - 0.487i)T + (-2.53 - 12.7i)T^{2} \)
17 \( 1 + (-2.54 + 1.05i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-7.35 + 0.724i)T + (18.6 - 3.70i)T^{2} \)
23 \( 1 + (-1.24 + 0.248i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.23 - 0.374i)T + (24.1 + 16.1i)T^{2} \)
31 \( 1 + (-5.60 - 5.60i)T + 31iT^{2} \)
37 \( 1 + (-0.0111 + 0.113i)T + (-36.2 - 7.21i)T^{2} \)
41 \( 1 + (-0.173 - 0.870i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-2.86 + 5.36i)T + (-23.8 - 35.7i)T^{2} \)
47 \( 1 + (1.87 + 4.52i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-2.71 + 0.823i)T + (44.0 - 29.4i)T^{2} \)
59 \( 1 + (5.87 + 7.16i)T + (-11.5 + 57.8i)T^{2} \)
61 \( 1 + (8.35 - 4.46i)T + (33.8 - 50.7i)T^{2} \)
67 \( 1 + (-13.4 + 7.19i)T + (37.2 - 55.7i)T^{2} \)
71 \( 1 + (0.0368 + 0.0550i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-2.32 - 1.55i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (1.80 - 4.35i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.75 + 17.8i)T + (-81.4 + 16.1i)T^{2} \)
89 \( 1 + (4.69 + 0.934i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-0.174 - 0.174i)T + 97iT^{2} \)
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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

−11.75385559361767960847625816806, −11.00168622612657435373611894296, −9.786896276304907135268242932254, −8.751553951958412298455648067101, −7.76989280373814359345775175524, −6.95764540193153710205798455705, −5.78348168797517398236146573023, −4.86719022296366960029658027108, −3.44249888665859231488216382265, −2.98976059196525146782723810316, 1.03241473035422244952636532643, 2.74961468162733180239107276593, 3.83911400989742089759801420539, 4.91982713728426196194362233692, 6.08284948590327569502489612902, 7.21951742566400011067469252266, 8.024216008435150082774785451957, 9.493527961685645227857327554824, 10.06948192918771526246861131835, 11.33342445486331275458832517740

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