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Properties

Label 2-96-32.29-c1-0-0
Degree $2$
Conductor $96$
Sign $-0.509 - 0.860i$
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  + (−0.126 + 1.40i)2-s + (−0.923 − 0.382i)3-s + (−1.96 − 0.355i)4-s + (1.36 + 3.28i)5-s + (0.655 − 1.25i)6-s + (−2.73 + 2.73i)7-s + (0.749 − 2.72i)8-s + (0.707 + 0.707i)9-s + (−4.80 + 1.50i)10-s + (3.01 − 1.24i)11-s + (1.68 + 1.08i)12-s + (0.932 − 2.25i)13-s + (−3.50 − 4.19i)14-s − 3.55i·15-s + (3.74 + 1.40i)16-s + 0.517i·17-s + ⋯
L(s)  = 1  + (−0.0893 + 0.996i)2-s + (−0.533 − 0.220i)3-s + (−0.984 − 0.177i)4-s + (0.609 + 1.47i)5-s + (0.267 − 0.511i)6-s + (−1.03 + 1.03i)7-s + (0.265 − 0.964i)8-s + (0.235 + 0.235i)9-s + (−1.51 + 0.475i)10-s + (0.909 − 0.376i)11-s + (0.485 + 0.312i)12-s + (0.258 − 0.624i)13-s + (−0.935 − 1.12i)14-s − 0.918i·15-s + (0.936 + 0.350i)16-s + 0.125i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.509 - 0.860i$
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: \((2,\ 96,\ (\ :1/2),\ -0.509 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381238 + 0.668754i\)
\(L(\frac12)\) \(\approx\) \(0.381238 + 0.668754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.126 - 1.40i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (-1.36 - 3.28i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.73 - 2.73i)T - 7iT^{2} \)
11 \( 1 + (-3.01 + 1.24i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.932 + 2.25i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.517iT - 17T^{2} \)
19 \( 1 + (-1.52 + 3.68i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.39 - 2.39i)T + 23iT^{2} \)
29 \( 1 + (-7.09 - 2.93i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.50T + 31T^{2} \)
37 \( 1 + (3.40 + 8.22i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.21 + 3.21i)T + 41iT^{2} \)
43 \( 1 + (1.31 - 0.544i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 4.67iT - 47T^{2} \)
53 \( 1 + (4.19 - 1.73i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (0.680 + 1.64i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-6.71 - 2.78i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-11.1 - 4.61i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-1.86 + 1.86i)T - 71iT^{2} \)
73 \( 1 + (9.06 + 9.06i)T + 73iT^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + (1.89 - 4.57i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.70 - 2.70i)T - 89iT^{2} \)
97 \( 1 - 3.73T + 97T^{2} \)
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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

−14.43242456717125310843433029157, −13.51095348892462551709048640171, −12.41902078291892647798990238634, −10.98168002094019898244761570313, −9.858123240664836154942439867512, −8.873689254640145893962169702396, −7.08695821385961858304760824810, −6.38117340404038053726656469901, −5.56693077946279116563294747801, −3.19719336357979669875244667282, 1.18348918079774660714853444571, 3.87891103785641849338781450797, 4.93621018443217669011619863659, 6.53884715052674029156479025308, 8.554621954609393656186645213568, 9.641941011584143531974865795888, 10.13363621339106314440146277538, 11.66618023406990955268891335249, 12.50205141130909614647437212168, 13.31252855878739734851045500542

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