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Properties

Label 2-96-96.11-c1-0-3
Degree $2$
Conductor $96$
Sign $-0.371 - 0.928i$
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  + (0.478 + 1.33i)2-s + (0.0499 + 1.73i)3-s + (−1.54 + 1.27i)4-s + (−1.06 − 0.443i)5-s + (−2.28 + 0.895i)6-s + (2.37 − 2.37i)7-s + (−2.43 − 1.44i)8-s + (−2.99 + 0.173i)9-s + (0.0777 − 1.63i)10-s + (5.50 + 2.27i)11-s + (−2.28 − 2.60i)12-s + (0.346 + 0.836i)13-s + (4.29 + 2.02i)14-s + (0.713 − 1.87i)15-s + (0.755 − 3.92i)16-s − 0.685·17-s + ⋯
L(s)  = 1  + (0.338 + 0.940i)2-s + (0.0288 + 0.999i)3-s + (−0.770 + 0.636i)4-s + (−0.478 − 0.198i)5-s + (−0.930 + 0.365i)6-s + (0.896 − 0.896i)7-s + (−0.860 − 0.509i)8-s + (−0.998 + 0.0576i)9-s + (0.0245 − 0.517i)10-s + (1.65 + 0.687i)11-s + (−0.658 − 0.752i)12-s + (0.0960 + 0.231i)13-s + (1.14 + 0.540i)14-s + (0.184 − 0.483i)15-s + (0.188 − 0.982i)16-s − 0.166·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.371 - 0.928i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1/2),\ -0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613277 + 0.905854i\)
\(L(\frac12)\) \(\approx\) \(0.613277 + 0.905854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.478 - 1.33i)T \)
3 \( 1 + (-0.0499 - 1.73i)T \)
good5 \( 1 + (1.06 + 0.443i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.37 + 2.37i)T - 7iT^{2} \)
11 \( 1 + (-5.50 - 2.27i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.346 - 0.836i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.685T + 17T^{2} \)
19 \( 1 + (3.35 - 1.38i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.05 + 2.05i)T - 23iT^{2} \)
29 \( 1 + (1.98 + 4.80i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 6.36iT - 31T^{2} \)
37 \( 1 + (-0.112 + 0.272i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.26 + 3.26i)T + 41iT^{2} \)
43 \( 1 + (0.993 - 2.39i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 + (1.56 - 3.77i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.20 - 5.33i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.53 - 0.636i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-4.69 - 11.3i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (7.99 + 7.99i)T + 71iT^{2} \)
73 \( 1 + (2.34 - 2.34i)T - 73iT^{2} \)
79 \( 1 - 8.91T + 79T^{2} \)
83 \( 1 + (3.06 + 7.40i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (1.99 - 1.99i)T - 89iT^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
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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

−14.59566518303251596328715364205, −13.72069790495763669968990462075, −12.15043289474720059331420361250, −11.19708933186701687135720631454, −9.737007876447935986806817998683, −8.693213818027413747184253862722, −7.60762100889187696916689079440, −6.21220541304989282234356545078, −4.39754738148223969339001216064, −4.13556893673780849481997255979, 1.68245916012246386026202753077, 3.45681632975919115294847020707, 5.30333280529911970510477963224, 6.63790618598279566234233279551, 8.389007662741675404035504675362, 9.050702855321277128855425758458, 11.04540888204867325450867935926, 11.62354020010215768817136557235, 12.34493135355524446856381459552, 13.52227671757622516338488987902

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