Properties

Label 2-384-16.13-c1-0-4
Degree $2$
Conductor $384$
Sign $0.997 + 0.0734i$
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  + (0.707 + 0.707i)3-s + (2.68 − 2.68i)5-s + 2.15i·7-s + 1.00i·9-s + (1.79 − 1.79i)11-s + (−1.38 − 1.38i)13-s + 3.79·15-s − 0.224·17-s + (0.158 + 0.158i)19-s + (−1.52 + 1.52i)21-s + 2.82i·23-s − 9.42i·25-s + (−0.707 + 0.707i)27-s + (1.85 + 1.85i)29-s − 1.84·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (1.20 − 1.20i)5-s + 0.816i·7-s + 0.333i·9-s + (0.542 − 0.542i)11-s + (−0.383 − 0.383i)13-s + 0.980·15-s − 0.0545·17-s + (0.0364 + 0.0364i)19-s + (−0.333 + 0.333i)21-s + 0.589i·23-s − 1.88i·25-s + (−0.136 + 0.136i)27-s + (0.344 + 0.344i)29-s − 0.330·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.997 + 0.0734i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.997 + 0.0734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85434 - 0.0681662i\)
\(L(\frac12)\) \(\approx\) \(1.85434 - 0.0681662i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (-2.68 + 2.68i)T - 5iT^{2} \)
7 \( 1 - 2.15iT - 7T^{2} \)
11 \( 1 + (-1.79 + 1.79i)T - 11iT^{2} \)
13 \( 1 + (1.38 + 1.38i)T + 13iT^{2} \)
17 \( 1 + 0.224T + 17T^{2} \)
19 \( 1 + (-0.158 - 0.158i)T + 19iT^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + (-1.85 - 1.85i)T + 29iT^{2} \)
31 \( 1 + 1.84T + 31T^{2} \)
37 \( 1 + (-3.66 + 3.66i)T - 37iT^{2} \)
41 \( 1 - 5.88iT - 41T^{2} \)
43 \( 1 + (7.75 - 7.75i)T - 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (7.51 - 7.51i)T - 53iT^{2} \)
59 \( 1 + (-4 + 4i)T - 59iT^{2} \)
61 \( 1 + (5.98 + 5.98i)T + 61iT^{2} \)
67 \( 1 + (10.4 + 10.4i)T + 67iT^{2} \)
71 \( 1 + 4.31iT - 71T^{2} \)
73 \( 1 - 5.97iT - 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 - 1.42iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−11.34998073909359633353750011336, −10.11016550078247058569692137676, −9.324335066616956916682669790956, −8.877468505005925184698661187041, −7.911800686163590900421768071830, −6.23413280662032687242420895974, −5.47060988726444034971261532509, −4.57303891753490048489946055690, −2.95247376561598072688580614664, −1.57892332046813321115411444929, 1.76995859892079450604564584527, 2.87644921536400208707403928719, 4.24033487173737568307061844510, 5.81352275730639343097652446999, 6.89633361370358691314290613492, 7.17165437259218619218218608944, 8.649973297996546057947774356623, 9.779437106439473751095207804553, 10.22272095291348009821922536243, 11.22492540242777672490825353850

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