Properties

Label 2-384-384.11-c1-0-23
Degree $2$
Conductor $384$
Sign $-0.542 - 0.840i$
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  + (1.02 + 0.974i)2-s + (−1.02 + 1.39i)3-s + (0.100 + 1.99i)4-s + (1.18 + 0.631i)5-s + (−2.41 + 0.429i)6-s + (4.72 + 0.940i)7-s + (−1.84 + 2.14i)8-s + (−0.893 − 2.86i)9-s + (0.595 + 1.79i)10-s + (2.00 − 1.64i)11-s + (−2.89 − 1.90i)12-s + (−2.66 + 1.42i)13-s + (3.92 + 5.57i)14-s + (−2.09 + 0.999i)15-s + (−3.97 + 0.402i)16-s + (−0.706 + 0.292i)17-s + ⋯
L(s)  = 1  + (0.724 + 0.689i)2-s + (−0.592 + 0.805i)3-s + (0.0504 + 0.998i)4-s + (0.528 + 0.282i)5-s + (−0.984 + 0.175i)6-s + (1.78 + 0.355i)7-s + (−0.651 + 0.758i)8-s + (−0.297 − 0.954i)9-s + (0.188 + 0.568i)10-s + (0.604 − 0.496i)11-s + (−0.834 − 0.551i)12-s + (−0.738 + 0.394i)13-s + (1.04 + 1.48i)14-s + (−0.540 + 0.258i)15-s + (−0.994 + 0.100i)16-s + (−0.171 + 0.0709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.542 - 0.840i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.542 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946568 + 1.73768i\)
\(L(\frac12)\) \(\approx\) \(0.946568 + 1.73768i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.02 - 0.974i)T \)
3 \( 1 + (1.02 - 1.39i)T \)
good5 \( 1 + (-1.18 - 0.631i)T + (2.77 + 4.15i)T^{2} \)
7 \( 1 + (-4.72 - 0.940i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-2.00 + 1.64i)T + (2.14 - 10.7i)T^{2} \)
13 \( 1 + (2.66 - 1.42i)T + (7.22 - 10.8i)T^{2} \)
17 \( 1 + (0.706 - 0.292i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-0.347 + 1.14i)T + (-15.7 - 10.5i)T^{2} \)
23 \( 1 + (7.23 + 4.83i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.806 + 0.661i)T + (5.65 + 28.4i)T^{2} \)
31 \( 1 + (-4.36 + 4.36i)T - 31iT^{2} \)
37 \( 1 + (-3.22 - 10.6i)T + (-30.7 + 20.5i)T^{2} \)
41 \( 1 + (0.972 + 0.649i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-0.495 + 5.03i)T + (-42.1 - 8.38i)T^{2} \)
47 \( 1 + (-6.98 + 2.89i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (7.86 - 6.45i)T + (10.3 - 51.9i)T^{2} \)
59 \( 1 + (-5.66 - 3.02i)T + (32.7 + 49.0i)T^{2} \)
61 \( 1 + (0.987 + 10.0i)T + (-59.8 + 11.9i)T^{2} \)
67 \( 1 + (7.67 - 0.756i)T + (65.7 - 13.0i)T^{2} \)
71 \( 1 + (-13.5 - 2.70i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (1.05 + 5.28i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-0.541 + 1.30i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (3.23 - 10.6i)T + (-69.0 - 46.1i)T^{2} \)
89 \( 1 + (0.415 + 0.621i)T + (-34.0 + 82.2i)T^{2} \)
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

−11.77158606047950225345583371581, −10.99256594979145570372109405178, −9.861073338725080237501670171627, −8.711061956384551850735676581908, −7.947407227570289724609628103224, −6.53744072071371245464384279943, −5.79845088211732210196798325888, −4.78345612065349487798416392853, −4.15711535693678631269929202170, −2.36622088067705518926352176465, 1.36137801912429231333050324435, 2.13785597483578980820654608270, 4.22700023910713767180468956873, 5.15489919091796708011107083569, 5.85051831656102594118740553336, 7.18895904957604691391825490721, 8.036529582091674205946098080943, 9.463470438151898381620682836501, 10.45767371119420665169544999852, 11.31870975449946093334431692990

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