Properties

Label 2-384-16.5-c1-0-7
Degree $2$
Conductor $384$
Sign $-0.489 + 0.872i$
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 + (−1.74 − 1.74i)5-s − 2.55i·7-s − 1.00i·9-s + (−0.473 − 0.473i)11-s + (−2.88 + 2.88i)13-s − 2.47·15-s − 6.44·17-s + (4.55 − 4.55i)19-s + (−1.80 − 1.80i)21-s − 2.82i·23-s + 1.11i·25-s + (−0.707 − 0.707i)27-s + (3.07 − 3.07i)29-s + 6.55·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.782 − 0.782i)5-s − 0.966i·7-s − 0.333i·9-s + (−0.142 − 0.142i)11-s + (−0.800 + 0.800i)13-s − 0.638·15-s − 1.56·17-s + (1.04 − 1.04i)19-s + (−0.394 − 0.394i)21-s − 0.589i·23-s + 0.223i·25-s + (−0.136 − 0.136i)27-s + (0.571 − 0.571i)29-s + 1.17·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.489 + 0.872i$
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: \((2,\ 384,\ (\ :1/2),\ -0.489 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554763 - 0.947467i\)
\(L(\frac12)\) \(\approx\) \(0.554763 - 0.947467i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 + 1.74i)T + 5iT^{2} \)
7 \( 1 + 2.55iT - 7T^{2} \)
11 \( 1 + (0.473 + 0.473i)T + 11iT^{2} \)
13 \( 1 + (2.88 - 2.88i)T - 13iT^{2} \)
17 \( 1 + 6.44T + 17T^{2} \)
19 \( 1 + (-4.55 + 4.55i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-3.07 + 3.07i)T - 29iT^{2} \)
31 \( 1 - 6.55T + 31T^{2} \)
37 \( 1 + (-2.72 - 2.72i)T + 37iT^{2} \)
41 \( 1 + 0.788iT - 41T^{2} \)
43 \( 1 + (-0.389 - 0.389i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-2.57 - 2.57i)T + 53iT^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + (-4.38 + 4.38i)T - 61iT^{2} \)
67 \( 1 + (-2.11 + 2.11i)T - 67iT^{2} \)
71 \( 1 - 5.11iT - 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + (0.641 - 0.641i)T - 83iT^{2} \)
89 \( 1 + 6.31iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−11.26702384626848174823621409163, −10.03151399958803264779986932673, −9.035705425662955314075390628320, −8.249497496892032419385869036901, −7.30296934575897060252286756377, −6.59371596483059161353430778057, −4.75341720893686343626649109457, −4.19431302167782812366478000999, −2.57687682205068795235215913770, −0.69554977315143057023148998299, 2.46333533971307305972152133665, 3.37946322201050372380329763652, 4.68620386841520595558449108363, 5.81707496981354703674105060840, 7.13347146357317806711047647142, 7.928709458902099522459473472430, 8.869983039364280552833611414353, 9.848876839383689034381056039947, 10.70134887308212841916371335782, 11.64736066246134242577157004255

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