Properties

Label 2-96-96.11-c1-0-8
Degree $2$
Conductor $96$
Sign $0.982 + 0.188i$
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  + (−0.838 + 1.13i)2-s + (0.602 − 1.62i)3-s + (−0.594 − 1.90i)4-s + (0.378 + 0.156i)5-s + (1.34 + 2.04i)6-s + (2.01 − 2.01i)7-s + (2.67 + 0.923i)8-s + (−2.27 − 1.95i)9-s + (−0.495 + 0.299i)10-s + (0.709 + 0.294i)11-s + (−3.45 − 0.184i)12-s + (2.08 + 5.03i)13-s + (0.604 + 3.97i)14-s + (0.482 − 0.520i)15-s + (−3.29 + 2.27i)16-s − 6.33·17-s + ⋯
L(s)  = 1  + (−0.592 + 0.805i)2-s + (0.347 − 0.937i)3-s + (−0.297 − 0.954i)4-s + (0.169 + 0.0701i)5-s + (0.549 + 0.835i)6-s + (0.760 − 0.760i)7-s + (0.945 + 0.326i)8-s + (−0.758 − 0.651i)9-s + (−0.156 + 0.0947i)10-s + (0.214 + 0.0886i)11-s + (−0.998 − 0.0531i)12-s + (0.577 + 1.39i)13-s + (0.161 + 1.06i)14-s + (0.124 − 0.134i)15-s + (−0.823 + 0.567i)16-s − 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.982 + 0.188i$
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: \((2,\ 96,\ (\ :1/2),\ 0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863931 - 0.0821120i\)
\(L(\frac12)\) \(\approx\) \(0.863931 - 0.0821120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.838 - 1.13i)T \)
3 \( 1 + (-0.602 + 1.62i)T \)
good5 \( 1 + (-0.378 - 0.156i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.01 + 2.01i)T - 7iT^{2} \)
11 \( 1 + (-0.709 - 0.294i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-2.08 - 5.03i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 6.33T + 17T^{2} \)
19 \( 1 + (-0.646 + 0.267i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.61 + 1.61i)T - 23iT^{2} \)
29 \( 1 + (-2.04 - 4.93i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 5.75iT - 31T^{2} \)
37 \( 1 + (2.50 - 6.04i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-5.52 - 5.52i)T + 41iT^{2} \)
43 \( 1 + (0.406 - 0.981i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 + (-0.674 + 1.62i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.35 - 8.08i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-4.14 + 1.71i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (2.65 + 6.40i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (1.97 + 1.97i)T + 71iT^{2} \)
73 \( 1 + (-9.48 + 9.48i)T - 73iT^{2} \)
79 \( 1 + 8.75T + 79T^{2} \)
83 \( 1 + (6.60 + 15.9i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.11 + 6.11i)T - 89iT^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
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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

−13.97978004349999800066412304334, −13.40606679714419247107129724912, −11.69410561369172323514034820322, −10.68714418776839619184078436453, −9.115987661598661650981331441748, −8.355768095921889073382635082783, −7.07257170039862236073717216823, −6.43179828628498079721096493295, −4.53732844009055038648679737649, −1.65454570551288944777479242008, 2.45736675395834477091544895829, 4.04545268443449136678427249039, 5.52016606072086687868540331403, 7.890041591130730399512216749963, 8.757625475447810928613050815710, 9.617940964594222592324754005106, 10.88078487179733717065127818810, 11.42134193846492222037782722507, 12.87538475691964391643655109031, 13.87949299907792288353330999409

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