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Properties

Label 2-384-48.35-c3-0-42
Degree $2$
Conductor $384$
Sign $-0.769 - 0.638i$
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  + (−3.76 − 3.58i)3-s + (−4.71 − 4.71i)5-s + 4.67·7-s + (1.36 + 26.9i)9-s + (29.7 − 29.7i)11-s + (−36.9 − 36.9i)13-s + (0.874 + 34.6i)15-s − 109. i·17-s + (−28.6 + 28.6i)19-s + (−17.6 − 16.7i)21-s + 0.193i·23-s − 80.5i·25-s + (91.4 − 106. i)27-s + (−162. + 162. i)29-s + 179. i·31-s + ⋯
L(s)  = 1  + (−0.724 − 0.689i)3-s + (−0.421 − 0.421i)5-s + 0.252·7-s + (0.0504 + 0.998i)9-s + (0.815 − 0.815i)11-s + (−0.788 − 0.788i)13-s + (0.0150 + 0.596i)15-s − 1.56i·17-s + (−0.345 + 0.345i)19-s + (−0.182 − 0.173i)21-s + 0.00175i·23-s − 0.644i·25-s + (0.651 − 0.758i)27-s + (−1.04 + 1.04i)29-s + 1.03i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.769 - 0.638i$
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: \((2,\ 384,\ (\ :3/2),\ -0.769 - 0.638i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3064405488\)
\(L(\frac12)\) \(\approx\) \(0.3064405488\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.76 + 3.58i)T \)
good5 \( 1 + (4.71 + 4.71i)T + 125iT^{2} \)
7 \( 1 - 4.67T + 343T^{2} \)
11 \( 1 + (-29.7 + 29.7i)T - 1.33e3iT^{2} \)
13 \( 1 + (36.9 + 36.9i)T + 2.19e3iT^{2} \)
17 \( 1 + 109. iT - 4.91e3T^{2} \)
19 \( 1 + (28.6 - 28.6i)T - 6.85e3iT^{2} \)
23 \( 1 - 0.193iT - 1.21e4T^{2} \)
29 \( 1 + (162. - 162. i)T - 2.43e4iT^{2} \)
31 \( 1 - 179. iT - 2.97e4T^{2} \)
37 \( 1 + (194. - 194. i)T - 5.06e4iT^{2} \)
41 \( 1 - 49.2T + 6.89e4T^{2} \)
43 \( 1 + (-336. - 336. i)T + 7.95e4iT^{2} \)
47 \( 1 + 187.T + 1.03e5T^{2} \)
53 \( 1 + (-195. - 195. i)T + 1.48e5iT^{2} \)
59 \( 1 + (302. - 302. i)T - 2.05e5iT^{2} \)
61 \( 1 + (501. + 501. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-36.4 + 36.4i)T - 3.00e5iT^{2} \)
71 \( 1 + 637. iT - 3.57e5T^{2} \)
73 \( 1 - 90.3iT - 3.89e5T^{2} \)
79 \( 1 - 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + (256. + 256. i)T + 5.71e5iT^{2} \)
89 \( 1 + 818.T + 7.04e5T^{2} \)
97 \( 1 - 667.T + 9.12e5T^{2} \)
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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

−10.62718867445615192656000684831, −9.388520846045782636452203113883, −8.326945542756350970824829356636, −7.49690953936051373365217161767, −6.57112286337409033826398656804, −5.43732504063001773525793857451, −4.62446919487736924066234086370, −2.99688307836753878403908456478, −1.30269405801400231887279520477, −0.12175216665445038328087599946, 1.89978235678384202999915999103, 3.82008911754305066084547374948, 4.38783973921696117641982495901, 5.69696370262413827947785608975, 6.70069732585728594690344288347, 7.56502393223438548640096286344, 8.981717323359892059646760552340, 9.707606341295795697425818354598, 10.66780735176451890676487792703, 11.41814954429915269863562478416

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