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Properties

Label 2-96-32.13-c1-0-2
Degree $2$
Conductor $96$
Sign $0.939 - 0.341i$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
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  + (−0.603 + 1.27i)2-s + (0.382 − 0.923i)3-s + (−1.27 − 1.54i)4-s + (3.68 − 1.52i)5-s + (0.950 + 1.04i)6-s + (−1.63 + 1.63i)7-s + (2.74 − 0.695i)8-s + (−0.707 − 0.707i)9-s + (−0.271 + 5.63i)10-s + (1.20 + 2.90i)11-s + (−1.91 + 0.584i)12-s + (−3.30 − 1.36i)13-s + (−1.10 − 3.08i)14-s − 3.99i·15-s + (−0.764 + 3.92i)16-s + 0.511i·17-s + ⋯
L(s)  = 1  + (−0.426 + 0.904i)2-s + (0.220 − 0.533i)3-s + (−0.635 − 0.771i)4-s + (1.64 − 0.683i)5-s + (0.388 + 0.427i)6-s + (−0.618 + 0.618i)7-s + (0.969 − 0.245i)8-s + (−0.235 − 0.235i)9-s + (−0.0858 + 1.78i)10-s + (0.363 + 0.876i)11-s + (−0.552 + 0.168i)12-s + (−0.915 − 0.379i)13-s + (−0.295 − 0.823i)14-s − 1.03i·15-s + (−0.191 + 0.981i)16-s + 0.124i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1/2),\ 0.939 - 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943796 + 0.166211i\)
\(L(\frac12)\) \(\approx\) \(0.943796 + 0.166211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.603 - 1.27i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (-3.68 + 1.52i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.63 - 1.63i)T - 7iT^{2} \)
11 \( 1 + (-1.20 - 2.90i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (3.30 + 1.36i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 0.511iT - 17T^{2} \)
19 \( 1 + (0.254 + 0.105i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.17 + 3.17i)T + 23iT^{2} \)
29 \( 1 + (2.75 - 6.64i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 5.82T + 31T^{2} \)
37 \( 1 + (5.53 - 2.29i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.94 - 3.94i)T + 41iT^{2} \)
43 \( 1 + (-1.30 - 3.15i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 9.41iT - 47T^{2} \)
53 \( 1 + (-2.50 - 6.04i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-13.3 + 5.52i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-3.35 + 8.10i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (1.11 - 2.69i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.26 + 1.26i)T - 71iT^{2} \)
73 \( 1 + (7.51 + 7.51i)T + 73iT^{2} \)
79 \( 1 + 0.709iT - 79T^{2} \)
83 \( 1 + (1.40 + 0.583i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.0856 - 0.0856i)T - 89iT^{2} \)
97 \( 1 + 0.677T + 97T^{2} \)
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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

−14.18745493249804544347043273595, −13.00457003506421497326284463086, −12.50324857832967735631028202867, −10.19125395118407617031274541523, −9.461903342151481831404377289344, −8.676182238988645816073897729151, −7.10777855284354623992236915371, −6.05967208001304641142178223571, −5.07886912835021564882845785024, −1.97305978887563522597829648323, 2.31269488456925567602490160136, 3.73612314798674563780303411620, 5.66431365875234917874445201565, 7.20185341768908409373154821000, 9.011952723680667503416768730958, 9.790982529657818156974111604366, 10.38660852621396502037206380932, 11.48018587227698318925332300387, 13.03680831786486802638400487011, 13.82976930843730096820454339318

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