Properties

Label 2-384-16.5-c1-0-2
Degree $2$
Conductor $384$
Sign $0.872 - 0.489i$
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 + (0.334 + 0.334i)5-s + 4.55i·7-s − 1.00i·9-s + (2.47 + 2.47i)11-s + (0.0594 − 0.0594i)13-s + 0.473·15-s + 3.61·17-s + (−2.55 + 2.55i)19-s + (3.22 + 3.22i)21-s − 2.82i·23-s − 4.77i·25-s + (−0.707 − 0.707i)27-s + (5.16 − 5.16i)29-s − 0.557·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.149 + 0.149i)5-s + 1.72i·7-s − 0.333i·9-s + (0.745 + 0.745i)11-s + (0.0164 − 0.0164i)13-s + 0.122·15-s + 0.877·17-s + (−0.586 + 0.586i)19-s + (0.703 + 0.703i)21-s − 0.589i·23-s − 0.955i·25-s + (−0.136 − 0.136i)27-s + (0.958 − 0.958i)29-s − 0.100·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.872 - 0.489i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.872 - 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56350 + 0.408721i\)
\(L(\frac12)\) \(\approx\) \(1.56350 + 0.408721i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.334 - 0.334i)T + 5iT^{2} \)
7 \( 1 - 4.55iT - 7T^{2} \)
11 \( 1 + (-2.47 - 2.47i)T + 11iT^{2} \)
13 \( 1 + (-0.0594 + 0.0594i)T - 13iT^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 + (2.55 - 2.55i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-5.16 + 5.16i)T - 29iT^{2} \)
31 \( 1 + 0.557T + 31T^{2} \)
37 \( 1 + (4.38 + 4.38i)T + 37iT^{2} \)
41 \( 1 - 9.27iT - 41T^{2} \)
43 \( 1 + (-1.61 - 1.61i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-0.493 - 0.493i)T + 53iT^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + (2.72 - 2.72i)T - 61iT^{2} \)
67 \( 1 + (3.77 - 3.77i)T - 67iT^{2} \)
71 \( 1 + 9.11iT - 71T^{2} \)
73 \( 1 - 0.541iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−11.84992528163165615995575595763, −10.34113250303791978491275644085, −9.453482378376616199073254043895, −8.651375874094775233556693767952, −7.85410705081533497271810187617, −6.51108940158708060113879518032, −5.85744070877236210604722731863, −4.47292682848476598746833523159, −2.91289612241523218119961145656, −1.89020047741944756437274000865, 1.20603024807497834306164677842, 3.30688755352605506775610469418, 4.07705397765460023138055660320, 5.27656954579541939383551991246, 6.69148333313821998587039406224, 7.49643283851883970303424838483, 8.583496898322148199759175567457, 9.455655516276754496600053485835, 10.45218349938692226375152381503, 10.95622444051876198825869411093

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