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Properties

Label 2-96-96.11-c1-0-11
Degree $2$
Conductor $96$
Sign $0.0514 + 0.998i$
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  + (0.310 − 1.37i)2-s + (0.988 − 1.42i)3-s + (−1.80 − 0.858i)4-s + (2.70 + 1.11i)5-s + (−1.65 − 1.80i)6-s + (−3.28 + 3.28i)7-s + (−1.74 + 2.22i)8-s + (−1.04 − 2.81i)9-s + (2.38 − 3.37i)10-s + (−0.276 − 0.114i)11-s + (−3.00 + 1.72i)12-s + (−0.155 − 0.374i)13-s + (3.50 + 5.54i)14-s + (4.26 − 2.73i)15-s + (2.52 + 3.10i)16-s + 1.41·17-s + ⋯
L(s)  = 1  + (0.219 − 0.975i)2-s + (0.570 − 0.821i)3-s + (−0.903 − 0.429i)4-s + (1.20 + 0.500i)5-s + (−0.675 − 0.737i)6-s + (−1.24 + 1.24i)7-s + (−0.617 + 0.786i)8-s + (−0.348 − 0.937i)9-s + (0.753 − 1.06i)10-s + (−0.0834 − 0.0345i)11-s + (−0.867 + 0.497i)12-s + (−0.0430 − 0.103i)13-s + (0.937 + 1.48i)14-s + (1.09 − 0.706i)15-s + (0.631 + 0.775i)16-s + 0.343·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0514 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0514 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.0514 + 0.998i$
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: \((2,\ 96,\ (\ :1/2),\ 0.0514 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891525 - 0.846813i\)
\(L(\frac12)\) \(\approx\) \(0.891525 - 0.846813i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.310 + 1.37i)T \)
3 \( 1 + (-0.988 + 1.42i)T \)
good5 \( 1 + (-2.70 - 1.11i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (3.28 - 3.28i)T - 7iT^{2} \)
11 \( 1 + (0.276 + 0.114i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.155 + 0.374i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + (2.51 - 1.04i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.30 + 3.30i)T - 23iT^{2} \)
29 \( 1 + (-0.650 - 1.57i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 4.16iT - 31T^{2} \)
37 \( 1 + (2.82 - 6.82i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.81 + 3.81i)T + 41iT^{2} \)
43 \( 1 + (-2.07 + 5.00i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 5.44iT - 47T^{2} \)
53 \( 1 + (-1.85 + 4.47i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.37 + 3.32i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (10.9 - 4.54i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-4.19 - 10.1i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (3.77 + 3.77i)T + 71iT^{2} \)
73 \( 1 + (3.89 - 3.89i)T - 73iT^{2} \)
79 \( 1 - 4.81T + 79T^{2} \)
83 \( 1 + (-3.60 - 8.70i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-3.69 + 3.69i)T - 89iT^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−13.41630566817982388288968960679, −12.77593806830471290277161392423, −11.95222402981521080868526922628, −10.33960321558283192918388440284, −9.448698946586838247391987046679, −8.623692235125312098877807194705, −6.53351478446303068522120710954, −5.68585839458761540965396583508, −3.12281476479600479654620754107, −2.21282182577892649635204461538, 3.43597493262163837737895274244, 4.80971058754842026650813240496, 6.11774919917092873302804255282, 7.40004443285738679360853486806, 8.936958032561368992737991933210, 9.654458362969393489901579776240, 10.45913448309824771171816968282, 12.80036727410939004603993140224, 13.51678269454705351493991686944, 14.07787337846566612948588079375

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