Properties

Label 2-384-384.11-c1-0-1
Degree $2$
Conductor $384$
Sign $-0.503 - 0.864i$
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  + (0.651 − 1.25i)2-s + (−1.52 + 0.818i)3-s + (−1.15 − 1.63i)4-s + (−0.995 − 0.532i)5-s + (0.0327 + 2.44i)6-s + (0.237 + 0.0473i)7-s + (−2.80 + 0.378i)8-s + (1.66 − 2.49i)9-s + (−1.31 + 0.902i)10-s + (−3.62 + 2.97i)11-s + (3.09 + 1.55i)12-s + (−2.36 + 1.26i)13-s + (0.214 − 0.267i)14-s + (1.95 − 0.00257i)15-s + (−1.35 + 3.76i)16-s + (−1.41 + 0.584i)17-s + ⋯
L(s)  = 1  + (0.460 − 0.887i)2-s + (−0.881 + 0.472i)3-s + (−0.575 − 0.817i)4-s + (−0.445 − 0.237i)5-s + (0.0133 + 0.999i)6-s + (0.0899 + 0.0178i)7-s + (−0.990 + 0.133i)8-s + (0.553 − 0.832i)9-s + (−0.416 + 0.285i)10-s + (−1.09 + 0.895i)11-s + (0.893 + 0.448i)12-s + (−0.655 + 0.350i)13-s + (0.0573 − 0.0715i)14-s + (0.504 − 0.000664i)15-s + (−0.337 + 0.941i)16-s + (−0.342 + 0.141i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.503 - 0.864i$
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: \((2,\ 384,\ (\ :1/2),\ -0.503 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0221911 + 0.0386067i\)
\(L(\frac12)\) \(\approx\) \(0.0221911 + 0.0386067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.651 + 1.25i)T \)
3 \( 1 + (1.52 - 0.818i)T \)
good5 \( 1 + (0.995 + 0.532i)T + (2.77 + 4.15i)T^{2} \)
7 \( 1 + (-0.237 - 0.0473i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (3.62 - 2.97i)T + (2.14 - 10.7i)T^{2} \)
13 \( 1 + (2.36 - 1.26i)T + (7.22 - 10.8i)T^{2} \)
17 \( 1 + (1.41 - 0.584i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (1.07 - 3.53i)T + (-15.7 - 10.5i)T^{2} \)
23 \( 1 + (2.30 + 1.54i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.329 + 0.270i)T + (5.65 + 28.4i)T^{2} \)
31 \( 1 + (2.76 - 2.76i)T - 31iT^{2} \)
37 \( 1 + (0.220 + 0.725i)T + (-30.7 + 20.5i)T^{2} \)
41 \( 1 + (0.703 + 0.470i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-0.314 + 3.18i)T + (-42.1 - 8.38i)T^{2} \)
47 \( 1 + (-6.23 + 2.58i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-5.81 + 4.77i)T + (10.3 - 51.9i)T^{2} \)
59 \( 1 + (5.39 + 2.88i)T + (32.7 + 49.0i)T^{2} \)
61 \( 1 + (1.29 + 13.1i)T + (-59.8 + 11.9i)T^{2} \)
67 \( 1 + (0.695 - 0.0684i)T + (65.7 - 13.0i)T^{2} \)
71 \( 1 + (5.49 + 1.09i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (1.31 + 6.59i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (2.49 - 6.02i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.21 - 13.8i)T + (-69.0 - 46.1i)T^{2} \)
89 \( 1 + (-8.98 - 13.4i)T + (-34.0 + 82.2i)T^{2} \)
97 \( 1 + (-3.38 - 3.38i)T + 97iT^{2} \)
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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

−11.77892832611976735197301375555, −10.69300844703902946212030777470, −10.18460773810775250463308695763, −9.317227339753124049943642438282, −8.010400642749729617305238185894, −6.66036828436612478639903873650, −5.44375831149199211857470043390, −4.68657155275461600890272675597, −3.82082487655025519229413048819, −2.10068442338898599925828382469, 0.02687052565797944464357165496, 2.82967305682595673347908015170, 4.38059787604915513988320458473, 5.38895769988971823630564505304, 6.11491143643312664840963783837, 7.37002293190819638117163265181, 7.71049046696515957290011256394, 8.909770337925193822367269702463, 10.30750164688511722999449865221, 11.27780892486302837819948065389

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