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Properties

Label 2-96-32.29-c1-0-4
Degree $2$
Conductor $96$
Sign $0.436 + 0.899i$
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  + (−0.947 + 1.04i)2-s + (−0.923 − 0.382i)3-s + (−0.204 − 1.98i)4-s + (−1.48 − 3.58i)5-s + (1.27 − 0.607i)6-s + (1.03 − 1.03i)7-s + (2.28 + 1.67i)8-s + (0.707 + 0.707i)9-s + (5.17 + 1.84i)10-s + (−2.98 + 1.23i)11-s + (−0.572 + 1.91i)12-s + (1.33 − 3.23i)13-s + (0.106 + 2.07i)14-s + 3.88i·15-s + (−3.91 + 0.811i)16-s − 5.31i·17-s + ⋯
L(s)  = 1  + (−0.670 + 0.742i)2-s + (−0.533 − 0.220i)3-s + (−0.102 − 0.994i)4-s + (−0.664 − 1.60i)5-s + (0.521 − 0.247i)6-s + (0.392 − 0.392i)7-s + (0.806 + 0.590i)8-s + (0.235 + 0.235i)9-s + (1.63 + 0.581i)10-s + (−0.901 + 0.373i)11-s + (−0.165 + 0.553i)12-s + (0.371 − 0.897i)13-s + (0.0283 + 0.554i)14-s + 1.00i·15-s + (−0.979 + 0.202i)16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1/2),\ 0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440888 - 0.276084i\)
\(L(\frac12)\) \(\approx\) \(0.440888 - 0.276084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.947 - 1.04i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (1.48 + 3.58i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.03 + 1.03i)T - 7iT^{2} \)
11 \( 1 + (2.98 - 1.23i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.33 + 3.23i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.31iT - 17T^{2} \)
19 \( 1 + (0.339 - 0.820i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.32 - 4.32i)T + 23iT^{2} \)
29 \( 1 + (-5.78 - 2.39i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 + (-0.646 - 1.56i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.42 - 3.42i)T + 41iT^{2} \)
43 \( 1 + (-6.50 + 2.69i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + (-6.63 + 2.74i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.185 + 0.447i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (3.16 + 1.31i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (6.09 + 2.52i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-2.91 + 2.91i)T - 71iT^{2} \)
73 \( 1 + (-1.02 - 1.02i)T + 73iT^{2} \)
79 \( 1 + 12.8iT - 79T^{2} \)
83 \( 1 + (6.41 - 15.4i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.991 - 0.991i)T - 89iT^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−13.67847540658918158146547971825, −12.80053216528761897854853059184, −11.60682650988557347461886604622, −10.46519222626004292801618308008, −9.154913853907573138362015942884, −8.069291042543239093168554002736, −7.32670004445895521072431975382, −5.43679888007340099731001788356, −4.75605404612409228711394227439, −0.867796908174544114716418541282, 2.68708545781212526151446259646, 4.16808256184701553277049876185, 6.35522244532724223516562220033, 7.56206070962654288206832131917, 8.720218940978036395718787765279, 10.35641318087286475163454526780, 10.89195731787729866963662050926, 11.59596534760119349995728684690, 12.74011636167362598208804193154, 14.20723009060344120256073287660

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