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Properties

Label 2-96-8.5-c3-0-4
Degree $2$
Conductor $96$
Sign $0.115 + 0.993i$
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 − 18.5i·5-s − 9.32·7-s − 9·9-s − 39.7i·11-s − 32.9i·13-s + 55.6·15-s + 90.5·17-s − 72.5i·19-s − 27.9i·21-s − 45.3·23-s − 218.·25-s − 27i·27-s + 143. i·29-s − 90.4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.65i·5-s − 0.503·7-s − 0.333·9-s − 1.08i·11-s − 0.703i·13-s + 0.957·15-s + 1.29·17-s − 0.876i·19-s − 0.290i·21-s − 0.411·23-s − 1.75·25-s − 0.192i·27-s + 0.918i·29-s − 0.524·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.115 + 0.993i$
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: \((2,\ 96,\ (\ :3/2),\ 0.115 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.935356 - 0.833110i\)
\(L(\frac12)\) \(\approx\) \(0.935356 - 0.833110i\)
\(L(\frac{5}{2})\) 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 \( 1 + 18.5iT - 125T^{2} \)
7 \( 1 + 9.32T + 343T^{2} \)
11 \( 1 + 39.7iT - 1.33e3T^{2} \)
13 \( 1 + 32.9iT - 2.19e3T^{2} \)
17 \( 1 - 90.5T + 4.91e3T^{2} \)
19 \( 1 + 72.5iT - 6.85e3T^{2} \)
23 \( 1 + 45.3T + 1.21e4T^{2} \)
29 \( 1 - 143. iT - 2.43e4T^{2} \)
31 \( 1 + 90.4T + 2.97e4T^{2} \)
37 \( 1 + 1.77iT - 5.06e4T^{2} \)
41 \( 1 - 195.T + 6.89e4T^{2} \)
43 \( 1 - 407. iT - 7.95e4T^{2} \)
47 \( 1 - 278.T + 1.03e5T^{2} \)
53 \( 1 + 241. iT - 1.48e5T^{2} \)
59 \( 1 + 149. iT - 2.05e5T^{2} \)
61 \( 1 - 508. iT - 2.26e5T^{2} \)
67 \( 1 + 950. iT - 3.00e5T^{2} \)
71 \( 1 - 803.T + 3.57e5T^{2} \)
73 \( 1 - 449.T + 3.89e5T^{2} \)
79 \( 1 - 157.T + 4.93e5T^{2} \)
83 \( 1 - 175. iT - 5.71e5T^{2} \)
89 \( 1 - 127.T + 7.04e5T^{2} \)
97 \( 1 - 158.T + 9.12e5T^{2} \)
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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

−13.08043695705294269752499190409, −12.35814218345204813913268143197, −11.08748808325813229555406744839, −9.767263424005604604493585002323, −8.903823842671997739587479984704, −7.947499564076599309192500612175, −5.87550081260865214497884497299, −4.94026200446015797201893645864, −3.40769092306494803031187260459, −0.74124780690162145530146580513, 2.21646132052290214849965656939, 3.68621033796480113681684864366, 5.94023095870860661389826426389, 6.96471856755003995010037164090, 7.74360824244540639999870215309, 9.634920169564429418080599254318, 10.43704125881227929661096221914, 11.69729902578726817228142255801, 12.55540094479460728814667984811, 13.97729224579363957454337446121

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