Properties

Label 2-384-384.11-c1-0-24
Degree $2$
Conductor $384$
Sign $0.789 - 0.614i$
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  + (−1.11 − 0.871i)2-s + (0.593 + 1.62i)3-s + (0.482 + 1.94i)4-s + (3.29 + 1.75i)5-s + (0.757 − 2.32i)6-s + (−0.370 − 0.0737i)7-s + (1.15 − 2.58i)8-s + (−2.29 + 1.93i)9-s + (−2.13 − 4.82i)10-s + (4.94 − 4.05i)11-s + (−2.87 + 1.93i)12-s + (2.85 − 1.52i)13-s + (0.348 + 0.405i)14-s + (−0.911 + 6.40i)15-s + (−3.53 + 1.87i)16-s + (−1.84 + 0.762i)17-s + ⋯
L(s)  = 1  + (−0.787 − 0.616i)2-s + (0.342 + 0.939i)3-s + (0.241 + 0.970i)4-s + (1.47 + 0.787i)5-s + (0.309 − 0.951i)6-s + (−0.140 − 0.0278i)7-s + (0.408 − 0.912i)8-s + (−0.765 + 0.643i)9-s + (−0.675 − 1.52i)10-s + (1.49 − 1.22i)11-s + (−0.829 + 0.558i)12-s + (0.791 − 0.422i)13-s + (0.0932 + 0.108i)14-s + (−0.235 + 1.65i)15-s + (−0.883 + 0.467i)16-s + (−0.446 + 0.184i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.789 - 0.614i$
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: \((2,\ 384,\ (\ :1/2),\ 0.789 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27947 + 0.439302i\)
\(L(\frac12)\) \(\approx\) \(1.27947 + 0.439302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 0.871i)T \)
3 \( 1 + (-0.593 - 1.62i)T \)
good5 \( 1 + (-3.29 - 1.75i)T + (2.77 + 4.15i)T^{2} \)
7 \( 1 + (0.370 + 0.0737i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-4.94 + 4.05i)T + (2.14 - 10.7i)T^{2} \)
13 \( 1 + (-2.85 + 1.52i)T + (7.22 - 10.8i)T^{2} \)
17 \( 1 + (1.84 - 0.762i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (1.60 - 5.28i)T + (-15.7 - 10.5i)T^{2} \)
23 \( 1 + (6.74 + 4.50i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-1.52 - 1.24i)T + (5.65 + 28.4i)T^{2} \)
31 \( 1 + (0.401 - 0.401i)T - 31iT^{2} \)
37 \( 1 + (-1.65 - 5.44i)T + (-30.7 + 20.5i)T^{2} \)
41 \( 1 + (0.603 + 0.403i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-0.837 + 8.50i)T + (-42.1 - 8.38i)T^{2} \)
47 \( 1 + (0.573 - 0.237i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-3.48 + 2.85i)T + (10.3 - 51.9i)T^{2} \)
59 \( 1 + (11.0 + 5.91i)T + (32.7 + 49.0i)T^{2} \)
61 \( 1 + (-0.569 - 5.78i)T + (-59.8 + 11.9i)T^{2} \)
67 \( 1 + (8.13 - 0.801i)T + (65.7 - 13.0i)T^{2} \)
71 \( 1 + (-3.18 - 0.633i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-0.359 - 1.80i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (1.60 - 3.87i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-0.559 + 1.84i)T + (-69.0 - 46.1i)T^{2} \)
89 \( 1 + (2.82 + 4.23i)T + (-34.0 + 82.2i)T^{2} \)
97 \( 1 + (-1.32 - 1.32i)T + 97iT^{2} \)
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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

−10.98084584127648167502995010652, −10.39406897714229093655240967572, −9.787358327638569282503080885289, −8.859426057169604768428232094378, −8.281164975188409417109833978092, −6.47937088660757718184602372923, −5.95260734709073011539150885648, −3.97354291564381546146302056544, −3.13393675495097061516584166691, −1.81966244602754516858379870973, 1.36874145803439441177738592409, 2.10747649733692410691674580158, 4.56305285949648076088308157035, 6.04280181657603685380802503858, 6.41849245526783434857835096802, 7.43667359814178522430155472486, 8.703236379638069395504136046804, 9.290478997259134078802304070562, 9.730543892680315466720929653427, 11.22312264442022455121913079047

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