Properties

Label 2-384-384.251-c1-0-13
Degree $2$
Conductor $384$
Sign $-0.931 - 0.362i$
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  + (−0.879 + 1.10i)2-s + (1.68 + 0.405i)3-s + (−0.452 − 1.94i)4-s + (−0.324 + 3.29i)5-s + (−1.93 + 1.50i)6-s + (−3.28 + 2.19i)7-s + (2.55 + 1.21i)8-s + (2.67 + 1.36i)9-s + (−3.36 − 3.25i)10-s + (−0.281 + 0.525i)11-s + (0.0294 − 3.46i)12-s + (−0.187 − 1.90i)13-s + (0.459 − 5.57i)14-s + (−1.88 + 5.41i)15-s + (−3.59 + 1.76i)16-s + (−2.42 − 5.84i)17-s + ⋯
L(s)  = 1  + (−0.622 + 0.782i)2-s + (0.972 + 0.234i)3-s + (−0.226 − 0.974i)4-s + (−0.145 + 1.47i)5-s + (−0.788 + 0.615i)6-s + (−1.24 + 0.829i)7-s + (0.903 + 0.428i)8-s + (0.890 + 0.455i)9-s + (−1.06 − 1.02i)10-s + (−0.0847 + 0.158i)11-s + (0.00850 − 0.999i)12-s + (−0.0521 − 0.529i)13-s + (0.122 − 1.48i)14-s + (−0.486 + 1.39i)15-s + (−0.897 + 0.440i)16-s + (−0.587 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.931 - 0.362i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.931 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188379 + 1.00387i\)
\(L(\frac12)\) \(\approx\) \(0.188379 + 1.00387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.879 - 1.10i)T \)
3 \( 1 + (-1.68 - 0.405i)T \)
good5 \( 1 + (0.324 - 3.29i)T + (-4.90 - 0.975i)T^{2} \)
7 \( 1 + (3.28 - 2.19i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.281 - 0.525i)T + (-6.11 - 9.14i)T^{2} \)
13 \( 1 + (0.187 + 1.90i)T + (-12.7 + 2.53i)T^{2} \)
17 \( 1 + (2.42 + 5.84i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (1.15 + 0.948i)T + (3.70 + 18.6i)T^{2} \)
23 \( 1 + (-1.22 - 6.16i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-3.15 - 5.90i)T + (-16.1 + 24.1i)T^{2} \)
31 \( 1 + (1.04 - 1.04i)T - 31iT^{2} \)
37 \( 1 + (-0.353 + 0.290i)T + (7.21 - 36.2i)T^{2} \)
41 \( 1 + (-2.35 - 11.8i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.626 - 2.06i)T + (-35.7 + 23.8i)T^{2} \)
47 \( 1 + (-0.749 - 1.80i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-5.62 + 10.5i)T + (-29.4 - 44.0i)T^{2} \)
59 \( 1 + (0.520 - 5.28i)T + (-57.8 - 11.5i)T^{2} \)
61 \( 1 + (-3.82 + 12.6i)T + (-50.7 - 33.8i)T^{2} \)
67 \( 1 + (-4.28 - 1.29i)T + (55.7 + 37.2i)T^{2} \)
71 \( 1 + (5.35 - 3.57i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-5.77 + 8.64i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (-11.1 - 4.62i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (2.60 + 2.13i)T + (16.1 + 81.4i)T^{2} \)
89 \( 1 + (-12.0 - 2.39i)T + (82.2 + 34.0i)T^{2} \)
97 \( 1 + (-5.23 - 5.23i)T + 97iT^{2} \)
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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

−11.40778065126405638093209706651, −10.46236703526673117870754968996, −9.628854596587129685544851830192, −9.157174811645703204661689918521, −7.950809544300454325801768632325, −7.03952922441722638550095631979, −6.48900325621783484955527982711, −5.10302809326445865864923552289, −3.32489174871415271784892748569, −2.51786046722952698824020318492, 0.74510024471982432717442324776, 2.25502750468278142770564017501, 3.82860580011027040414839585378, 4.31270718728740627971655006707, 6.44374821449545990343457161082, 7.53674500844144473161080121018, 8.608567690761886059710218464036, 8.910511639525011726309971118853, 9.931885068688423413436014192309, 10.61419937198326611884223847051

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