Properties

Label 2-384-384.227-c1-0-3
Degree $2$
Conductor $384$
Sign $-0.694 - 0.719i$
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  + (−0.617 − 1.27i)2-s + (−0.948 + 1.44i)3-s + (−1.23 + 1.57i)4-s + (0.857 + 1.60i)5-s + (2.42 + 0.311i)6-s + (−1.64 + 0.327i)7-s + (2.76 + 0.604i)8-s + (−1.20 − 2.74i)9-s + (1.51 − 2.08i)10-s + (2.14 − 2.61i)11-s + (−1.10 − 3.28i)12-s + (−2.50 + 4.68i)13-s + (1.43 + 1.89i)14-s + (−3.13 − 0.278i)15-s + (−0.937 − 3.88i)16-s + (−1.09 − 0.454i)17-s + ⋯
L(s)  = 1  + (−0.436 − 0.899i)2-s + (−0.547 + 0.836i)3-s + (−0.618 + 0.785i)4-s + (0.383 + 0.717i)5-s + (0.991 + 0.127i)6-s + (−0.621 + 0.123i)7-s + (0.976 + 0.213i)8-s + (−0.400 − 0.916i)9-s + (0.478 − 0.658i)10-s + (0.645 − 0.786i)11-s + (−0.318 − 0.947i)12-s + (−0.694 + 1.29i)13-s + (0.382 + 0.505i)14-s + (−0.810 − 0.0718i)15-s + (−0.234 − 0.972i)16-s + (−0.266 − 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.694 - 0.719i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143720 + 0.338404i\)
\(L(\frac12)\) \(\approx\) \(0.143720 + 0.338404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.617 + 1.27i)T \)
3 \( 1 + (0.948 - 1.44i)T \)
good5 \( 1 + (-0.857 - 1.60i)T + (-2.77 + 4.15i)T^{2} \)
7 \( 1 + (1.64 - 0.327i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-2.14 + 2.61i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (2.50 - 4.68i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (1.09 + 0.454i)T + (12.0 + 12.0i)T^{2} \)
19 \( 1 + (8.04 - 2.44i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (6.42 - 4.29i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.18 - 1.44i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (5.85 + 5.85i)T + 31iT^{2} \)
37 \( 1 + (-6.63 - 2.01i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (3.28 - 2.19i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (8.90 - 0.877i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (-7.69 - 3.18i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (-2.88 + 3.51i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (0.303 + 0.567i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (-7.67 - 0.755i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (0.0194 - 0.197i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (6.77 - 1.34i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (0.818 - 4.11i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-2.06 - 4.97i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-15.1 + 4.60i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (0.0260 - 0.0390i)T + (-34.0 - 82.2i)T^{2} \)
97 \( 1 + (11.2 - 11.2i)T - 97iT^{2} \)
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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

−11.49729253465343537611970148222, −10.74710788190099816366358090831, −9.877484890126932407163297905669, −9.375669455253227076461341289568, −8.387863965761934292213847114519, −6.76655151551732808885885333499, −6.00611398947101018099467337934, −4.38774896963087294370163920993, −3.60476080762848455192663048180, −2.21333165997250472200578346842, 0.28769814768252586531238596753, 1.99201024221645030284192284828, 4.44842614298090773343697785104, 5.43254415727562681672248547351, 6.38889301638974513384964772241, 7.05719941817561575885757201468, 8.150896323013957849211035360311, 8.931695668389309891151813313788, 10.04785375038206726255469308562, 10.69737855456097420508608665129

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