Properties

Label 2-384-48.35-c1-0-5
Degree $2$
Conductor $384$
Sign $0.958 - 0.283i$
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  + (1.43 + 0.966i)3-s + (−1.57 − 1.57i)5-s + 2.24·7-s + (1.13 + 2.77i)9-s + (1.13 − 1.13i)11-s + (3.24 + 3.24i)13-s + (−0.739 − 3.77i)15-s + 1.66i·17-s + (3.77 − 3.77i)19-s + (3.23 + 2.17i)21-s − 2.26i·23-s − 0.0586i·25-s + (−1.05 + 5.08i)27-s + (−3.23 + 3.23i)29-s + 1.30i·31-s + ⋯
L(s)  = 1  + (0.829 + 0.558i)3-s + (−0.702 − 0.702i)5-s + 0.850·7-s + (0.377 + 0.926i)9-s + (0.341 − 0.341i)11-s + (0.901 + 0.901i)13-s + (−0.191 − 0.975i)15-s + 0.403i·17-s + (0.866 − 0.866i)19-s + (0.705 + 0.474i)21-s − 0.471i·23-s − 0.0117i·25-s + (−0.203 + 0.978i)27-s + (−0.600 + 0.600i)29-s + 0.234i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.958 - 0.283i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.958 - 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77222 + 0.256512i\)
\(L(\frac12)\) \(\approx\) \(1.77222 + 0.256512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 - 0.966i)T \)
good5 \( 1 + (1.57 + 1.57i)T + 5iT^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
11 \( 1 + (-1.13 + 1.13i)T - 11iT^{2} \)
13 \( 1 + (-3.24 - 3.24i)T + 13iT^{2} \)
17 \( 1 - 1.66iT - 17T^{2} \)
19 \( 1 + (-3.77 + 3.77i)T - 19iT^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 + (3.23 - 3.23i)T - 29iT^{2} \)
31 \( 1 - 1.30iT - 31T^{2} \)
37 \( 1 + (2.30 - 2.30i)T - 37iT^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + (3.77 + 3.77i)T + 43iT^{2} \)
47 \( 1 - 3.74T + 47T^{2} \)
53 \( 1 + (-0.972 - 0.972i)T + 53iT^{2} \)
59 \( 1 + (-3.88 + 3.88i)T - 59iT^{2} \)
61 \( 1 + (4.19 + 4.19i)T + 61iT^{2} \)
67 \( 1 + (8.02 - 8.02i)T - 67iT^{2} \)
71 \( 1 + 11.0iT - 71T^{2} \)
73 \( 1 + 6.38iT - 73T^{2} \)
79 \( 1 - 2.69iT - 79T^{2} \)
83 \( 1 + (-2.61 - 2.61i)T + 83iT^{2} \)
89 \( 1 - 7.35T + 89T^{2} \)
97 \( 1 + 5.67T + 97T^{2} \)
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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

−11.36306195365863266996232987427, −10.53418882412357062752171482849, −9.225265792680689010892921455752, −8.638992573968847568387171914651, −8.010225554601660588847404675163, −6.83548184431543851711572588117, −5.17630022367361949852725778255, −4.35330730347338546020520073998, −3.40432520046975422127802405606, −1.61553805336370046337646677342, 1.51842017909992403566791883762, 3.11146926724121370900706633168, 3.94829062619089678698972476093, 5.55481990963952245572096158220, 6.86761590404151259386627950781, 7.71553708461028661393037993999, 8.215153551137353558791564325032, 9.357293956080246625269106681041, 10.41947040954832501048856563096, 11.48564821596018514765484924344

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