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Properties

Label 2-96-8.5-c3-0-2
Degree $2$
Conductor $96$
Sign $0.427 - 0.903i$
Analytic cond. $5.66418$
Root an. cond. $2.37995$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 0.612i·5-s + 22.7·7-s − 9·9-s + 60.2i·11-s + 52.9i·13-s + 1.83·15-s + 47.1·17-s − 29.1i·19-s + 68.2i·21-s − 109.·23-s + 124.·25-s − 27i·27-s − 10.4i·29-s + 220.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.0547i·5-s + 1.22·7-s − 0.333·9-s + 1.65i·11-s + 1.12i·13-s + 0.0316·15-s + 0.672·17-s − 0.352i·19-s + 0.709i·21-s − 0.992·23-s + 0.996·25-s − 0.192i·27-s − 0.0667i·29-s + 1.27·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.427 - 0.903i$
Analytic conductor: \(5.66418\)
Root analytic conductor: \(2.37995\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :3/2),\ 0.427 - 0.903i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.40024 + 0.886509i\)
\(L(\frac12)\) \(\approx\) \(1.40024 + 0.886509i\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
good5 \( 1 + 0.612iT - 125T^{2} \)
7 \( 1 - 22.7T + 343T^{2} \)
11 \( 1 - 60.2iT - 1.33e3T^{2} \)
13 \( 1 - 52.9iT - 2.19e3T^{2} \)
17 \( 1 - 47.1T + 4.91e3T^{2} \)
19 \( 1 + 29.1iT - 6.85e3T^{2} \)
23 \( 1 + 109.T + 1.21e4T^{2} \)
29 \( 1 + 10.4iT - 2.43e4T^{2} \)
31 \( 1 - 220.T + 2.97e4T^{2} \)
37 \( 1 + 408. iT - 5.06e4T^{2} \)
41 \( 1 + 360.T + 6.89e4T^{2} \)
43 \( 1 + 236. iT - 7.95e4T^{2} \)
47 \( 1 + 129.T + 1.03e5T^{2} \)
53 \( 1 + 117. iT - 1.48e5T^{2} \)
59 \( 1 + 262. iT - 2.05e5T^{2} \)
61 \( 1 - 273. iT - 2.26e5T^{2} \)
67 \( 1 + 89.4iT - 3.00e5T^{2} \)
71 \( 1 - 350.T + 3.57e5T^{2} \)
73 \( 1 - 532.T + 3.89e5T^{2} \)
79 \( 1 - 166.T + 4.93e5T^{2} \)
83 \( 1 - 361. iT - 5.71e5T^{2} \)
89 \( 1 - 40.3T + 7.04e5T^{2} \)
97 \( 1 + 614.T + 9.12e5T^{2} \)
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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

−13.99196963554436405536025241066, −12.37128819844717534980187252489, −11.57905096378395016030190116622, −10.39166112859402680759411472612, −9.393208089701774574074209121972, −8.163604944758821765782660197407, −6.92620319782875688718453191580, −5.10358281834891909894618662917, −4.22645757138495605512171968302, −1.96172237674734729345538275186, 1.09999676689116059091841406697, 3.12210026545588998535672609522, 5.12778777022101366021161096721, 6.28294529778078933220065088040, 8.037388212965681409055556069259, 8.333847521904450220817807060741, 10.24235431740687672921695475086, 11.27171268945163775357138239647, 12.13128679009365501666463664687, 13.44311101427324408421837291811

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