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Properties

Label 2-384-128.29-c1-0-2
Degree $2$
Conductor $384$
Sign $-0.174 - 0.984i$
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  + (−0.706 − 1.22i)2-s + (−0.0980 + 0.995i)3-s + (−1.00 + 1.73i)4-s + (−0.708 + 0.378i)5-s + (1.28 − 0.582i)6-s + (−0.234 − 1.17i)7-s + (2.82 + 0.00532i)8-s + (−0.980 − 0.195i)9-s + (0.963 + 0.600i)10-s + (−2.55 + 3.11i)11-s + (−1.62 − 1.16i)12-s + (1.09 − 2.05i)13-s + (−1.27 + 1.12i)14-s + (−0.307 − 0.741i)15-s + (−1.99 − 3.46i)16-s + (−2.92 + 7.07i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−0.0565 + 0.574i)3-s + (−0.501 + 0.865i)4-s + (−0.316 + 0.169i)5-s + (0.526 − 0.237i)6-s + (−0.0886 − 0.445i)7-s + (0.999 + 0.00188i)8-s + (−0.326 − 0.0650i)9-s + (0.304 + 0.189i)10-s + (−0.771 + 0.939i)11-s + (−0.468 − 0.336i)12-s + (0.304 − 0.570i)13-s + (−0.341 + 0.299i)14-s + (−0.0793 − 0.191i)15-s + (−0.497 − 0.867i)16-s + (−0.710 + 1.71i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.174 - 0.984i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.174 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303926 + 0.362595i\)
\(L(\frac12)\) \(\approx\) \(0.303926 + 0.362595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.706 + 1.22i)T \)
3 \( 1 + (0.0980 - 0.995i)T \)
good5 \( 1 + (0.708 - 0.378i)T + (2.77 - 4.15i)T^{2} \)
7 \( 1 + (0.234 + 1.17i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (2.55 - 3.11i)T + (-2.14 - 10.7i)T^{2} \)
13 \( 1 + (-1.09 + 2.05i)T + (-7.22 - 10.8i)T^{2} \)
17 \( 1 + (2.92 - 7.07i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.0109 + 0.0361i)T + (-15.7 + 10.5i)T^{2} \)
23 \( 1 + (6.86 - 4.58i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-2.83 + 2.32i)T + (5.65 - 28.4i)T^{2} \)
31 \( 1 + (6.28 - 6.28i)T - 31iT^{2} \)
37 \( 1 + (-3.55 - 1.07i)T + (30.7 + 20.5i)T^{2} \)
41 \( 1 + (4.73 + 7.08i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.229 + 2.33i)T + (-42.1 + 8.38i)T^{2} \)
47 \( 1 + (-7.57 - 3.13i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (0.642 + 0.526i)T + (10.3 + 51.9i)T^{2} \)
59 \( 1 + (-1.13 - 2.12i)T + (-32.7 + 49.0i)T^{2} \)
61 \( 1 + (-11.4 - 1.12i)T + (59.8 + 11.9i)T^{2} \)
67 \( 1 + (9.62 + 0.947i)T + (65.7 + 13.0i)T^{2} \)
71 \( 1 + (4.05 - 0.806i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-1.41 + 7.09i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-3.50 + 1.45i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (14.3 - 4.35i)T + (69.0 - 46.1i)T^{2} \)
89 \( 1 + (-3.20 - 2.14i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (5.54 - 5.54i)T - 97iT^{2} \)
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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

−11.36937355668105052615536485677, −10.35707889929237902934332024026, −10.25420473063795806018457455807, −8.979032263319228578055362119545, −8.061053032445983153743083845491, −7.23865293381895094344416278087, −5.61501732519605076557085780809, −4.24585342622572014086193317510, −3.54578411150347364947076444271, −1.97494266300422428600338014629, 0.36038258260596658542810667940, 2.40786088696547923559929094525, 4.36251959554239692062700336483, 5.57487977532108740749627707071, 6.37559976058645616650928924833, 7.39557442004262249361417711100, 8.271211177974707088263945437300, 8.923772137838539963592906787683, 9.970538531915999617495944038594, 11.12370106189311770137690734338

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