Properties

Label 2-384-8.5-c7-0-37
Degree $2$
Conductor $384$
Sign $-0.707 + 0.707i$
Analytic cond. $119.955$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27i·3-s + 135. i·5-s − 678.·7-s − 729·9-s + 5.73e3i·11-s + 7.35e3i·13-s + 3.67e3·15-s − 1.55e4·17-s + 8.42e3i·19-s + 1.83e4i·21-s − 3.46e4·23-s + 5.96e4·25-s + 1.96e4i·27-s + 1.33e5i·29-s − 5.44e3·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.486i·5-s − 0.747·7-s − 0.333·9-s + 1.29i·11-s + 0.928i·13-s + 0.280·15-s − 0.768·17-s + 0.281i·19-s + 0.431i·21-s − 0.594·23-s + 0.763·25-s + 0.192i·27-s + 1.01i·29-s − 0.0328·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(119.955\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :7/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1385418295\)
\(L(\frac12)\) \(\approx\) \(0.1385418295\)
\(L(\frac{9}{2})\) 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 \( 1 + 27iT \)
good5 \( 1 - 135. iT - 7.81e4T^{2} \)
7 \( 1 + 678.T + 8.23e5T^{2} \)
11 \( 1 - 5.73e3iT - 1.94e7T^{2} \)
13 \( 1 - 7.35e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.55e4T + 4.10e8T^{2} \)
19 \( 1 - 8.42e3iT - 8.93e8T^{2} \)
23 \( 1 + 3.46e4T + 3.40e9T^{2} \)
29 \( 1 - 1.33e5iT - 1.72e10T^{2} \)
31 \( 1 + 5.44e3T + 2.75e10T^{2} \)
37 \( 1 + 4.16e5iT - 9.49e10T^{2} \)
41 \( 1 - 4.32e5T + 1.94e11T^{2} \)
43 \( 1 - 4.10e5iT - 2.71e11T^{2} \)
47 \( 1 + 8.59e5T + 5.06e11T^{2} \)
53 \( 1 + 7.56e5iT - 1.17e12T^{2} \)
59 \( 1 + 2.13e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.96e5iT - 3.14e12T^{2} \)
67 \( 1 - 2.17e6iT - 6.06e12T^{2} \)
71 \( 1 - 5.94e5T + 9.09e12T^{2} \)
73 \( 1 - 1.08e6T + 1.10e13T^{2} \)
79 \( 1 - 1.51e6T + 1.92e13T^{2} \)
83 \( 1 + 3.23e6iT - 2.71e13T^{2} \)
89 \( 1 - 4.95e6T + 4.42e13T^{2} \)
97 \( 1 - 1.35e7T + 8.07e13T^{2} \)
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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

−9.731726765570175878071527348363, −9.018043238539180861108685876100, −7.74518780669912293077602547954, −6.83255938693116673563281247397, −6.39700651961219750888380903605, −4.93368581753322454341739170919, −3.78527923492959787499853336775, −2.52471891590517805469820950729, −1.63557579436524281887523390688, −0.03414949023601577846083792027, 0.867955422876507878198037407735, 2.66367915932821032618084381600, 3.53733364759281153737703917591, 4.67165787786052235599270510029, 5.73154048648330209246744022473, 6.50790148092949281346898848379, 7.977859802004594254490150936273, 8.722601543663427810437077387349, 9.561582192490893640007817924204, 10.47524062779063983844442847057

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