Properties

Label 2-384-128.85-c1-0-14
Degree $2$
Conductor $384$
Sign $0.751 - 0.660i$
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  + (1.23 + 0.686i)2-s + (0.634 − 0.773i)3-s + (1.05 + 1.69i)4-s + (−1.57 + 0.478i)5-s + (1.31 − 0.519i)6-s + (1.01 + 0.202i)7-s + (0.139 + 2.82i)8-s + (−0.195 − 0.980i)9-s + (−2.28 − 0.492i)10-s + (6.16 + 0.607i)11-s + (1.98 + 0.260i)12-s + (0.803 − 2.64i)13-s + (1.11 + 0.948i)14-s + (−0.631 + 1.52i)15-s + (−1.76 + 3.58i)16-s + (1.52 + 3.66i)17-s + ⋯
L(s)  = 1  + (0.874 + 0.485i)2-s + (0.366 − 0.446i)3-s + (0.528 + 0.849i)4-s + (−0.706 + 0.214i)5-s + (0.536 − 0.212i)6-s + (0.384 + 0.0764i)7-s + (0.0494 + 0.998i)8-s + (−0.0650 − 0.326i)9-s + (−0.721 − 0.155i)10-s + (1.85 + 0.183i)11-s + (0.572 + 0.0752i)12-s + (0.222 − 0.734i)13-s + (0.299 + 0.253i)14-s + (−0.163 + 0.393i)15-s + (−0.441 + 0.897i)16-s + (0.368 + 0.890i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.751 - 0.660i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.751 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25575 + 0.850225i\)
\(L(\frac12)\) \(\approx\) \(2.25575 + 0.850225i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 - 0.686i)T \)
3 \( 1 + (-0.634 + 0.773i)T \)
good5 \( 1 + (1.57 - 0.478i)T + (4.15 - 2.77i)T^{2} \)
7 \( 1 + (-1.01 - 0.202i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (-6.16 - 0.607i)T + (10.7 + 2.14i)T^{2} \)
13 \( 1 + (-0.803 + 2.64i)T + (-10.8 - 7.22i)T^{2} \)
17 \( 1 + (-1.52 - 3.66i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (6.52 - 3.48i)T + (10.5 - 15.7i)T^{2} \)
23 \( 1 + (-1.64 + 2.45i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (0.794 + 8.06i)T + (-28.4 + 5.65i)T^{2} \)
31 \( 1 + (7.54 + 7.54i)T + 31iT^{2} \)
37 \( 1 + (-3.34 + 6.26i)T + (-20.5 - 30.7i)T^{2} \)
41 \( 1 + (-5.94 - 3.96i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.180 + 0.219i)T + (-8.38 + 42.1i)T^{2} \)
47 \( 1 + (6.91 - 2.86i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (0.0320 - 0.325i)T + (-51.9 - 10.3i)T^{2} \)
59 \( 1 + (4.12 + 13.6i)T + (-49.0 + 32.7i)T^{2} \)
61 \( 1 + (-4.58 - 3.76i)T + (11.9 + 59.8i)T^{2} \)
67 \( 1 + (4.00 + 3.28i)T + (13.0 + 65.7i)T^{2} \)
71 \( 1 + (1.36 - 6.88i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (1.14 - 0.228i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-9.75 - 4.04i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-2.86 - 5.36i)T + (-46.1 + 69.0i)T^{2} \)
89 \( 1 + (-6.07 - 9.08i)T + (-34.0 + 82.2i)T^{2} \)
97 \( 1 + (2.11 + 2.11i)T + 97iT^{2} \)
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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

−11.59130366753234352798266399246, −10.99275534343104462017125777283, −9.391957532194177899568094957845, −8.183137169287322185226001739606, −7.78861223505128520799180803625, −6.54082997553332489932809863293, −5.91776413790409712332821876958, −4.15969685750127283854252592307, −3.71330872670493181988451440729, −1.99132557388459156384839356864, 1.57308740689897772235688114321, 3.31218006339621414712145284315, 4.16719445027787094350102331782, 4.92572189764147349719263001992, 6.41325609543091893282526551197, 7.25822091283288607119380672981, 8.820444856848170444140730915363, 9.292889977410561062835363585929, 10.65631331246293232640966435893, 11.41308474148834255462717780412

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