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Properties

Label 2-384-24.11-c5-0-44
Degree $2$
Conductor $384$
Sign $i$
Analytic cond. $61.5873$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.5i·3-s − 31.1·5-s + 93.0i·7-s − 243·9-s + 308. i·11-s + 484. i·15-s + 1.45e3·21-s − 2.15e3·25-s + 3.78e3i·27-s + 8.79e3·29-s − 8.13e3i·31-s + 4.80e3·33-s − 2.89e3i·35-s + 7.56e3·45-s + 8.14e3·49-s + ⋯
L(s)  = 1  − 0.999i·3-s − 0.556·5-s + 0.717i·7-s − 9-s + 0.768i·11-s + 0.556i·15-s + 0.717·21-s − 0.690·25-s + 1.00i·27-s + 1.94·29-s − 1.52i·31-s + 0.768·33-s − 0.399i·35-s + 0.556·45-s + 0.484·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $i$
Analytic conductor: \(61.5873\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :5/2),\ i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.317906853\)
\(L(\frac12)\) \(\approx\) \(1.317906853\)
\(L(\frac{7}{2})\) 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 \( 1 + 15.5iT \)
good5 \( 1 + 31.1T + 3.12e3T^{2} \)
7 \( 1 - 93.0iT - 1.68e4T^{2} \)
11 \( 1 - 308. iT - 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 - 1.41e6T^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 + 6.43e6T^{2} \)
29 \( 1 - 8.79e3T + 2.05e7T^{2} \)
31 \( 1 + 8.13e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 - 1.15e8T^{2} \)
43 \( 1 + 1.47e8T^{2} \)
47 \( 1 + 2.29e8T^{2} \)
53 \( 1 - 1.47e4T + 4.18e8T^{2} \)
59 \( 1 + 2.90e4iT - 7.14e8T^{2} \)
61 \( 1 - 8.44e8T^{2} \)
67 \( 1 + 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 + 9.07e4T + 2.07e9T^{2} \)
79 \( 1 + 1.09e5iT - 3.07e9T^{2} \)
83 \( 1 + 952. iT - 3.93e9T^{2} \)
89 \( 1 - 5.58e9T^{2} \)
97 \( 1 - 9.02e4T + 8.58e9T^{2} \)
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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

−10.33186642587339294975988137938, −9.195890570205527713988092195815, −8.258517068715227616701726653423, −7.54143170561936692896544448039, −6.56176616190729582609386688324, −5.62391088519168724538850193133, −4.35743311633985370324555631004, −2.87689873501416663878568012313, −1.87181747860969257149205965894, −0.43515881772642043979118623060, 0.856505901096000704017256424298, 2.88782858940028206805772487375, 3.83479861656503842065609789313, 4.68466801602957303722782278440, 5.82285711079206753399629935515, 7.00469677569301500266985583553, 8.194463125823097302363010148433, 8.840270251268702448006149534604, 10.06822812907990013536276567441, 10.61207422399423165262490750934

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