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Properties

Label 2-96-32.13-c1-0-7
Degree $2$
Conductor $96$
Sign $-0.207 + 0.978i$
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  + (0.333 − 1.37i)2-s + (0.382 − 0.923i)3-s + (−1.77 − 0.916i)4-s + (−1.20 + 0.498i)5-s + (−1.14 − 0.834i)6-s + (2.59 − 2.59i)7-s + (−1.85 + 2.13i)8-s + (−0.707 − 0.707i)9-s + (0.283 + 1.82i)10-s + (2.14 + 5.18i)11-s + (−1.52 + 1.29i)12-s + (−0.984 − 0.407i)13-s + (−2.69 − 4.43i)14-s + 1.30i·15-s + (2.31 + 3.25i)16-s − 0.979i·17-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)2-s + (0.220 − 0.533i)3-s + (−0.888 − 0.458i)4-s + (−0.538 + 0.223i)5-s + (−0.466 − 0.340i)6-s + (0.980 − 0.980i)7-s + (−0.655 + 0.755i)8-s + (−0.235 − 0.235i)9-s + (0.0897 + 0.575i)10-s + (0.647 + 1.56i)11-s + (−0.440 + 0.372i)12-s + (−0.272 − 0.113i)13-s + (−0.721 − 1.18i)14-s + 0.336i·15-s + (0.579 + 0.814i)16-s − 0.237i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.207 + 0.978i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1/2),\ -0.207 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681599 - 0.841238i\)
\(L(\frac12)\) \(\approx\) \(0.681599 - 0.841238i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.333 + 1.37i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (1.20 - 0.498i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.59 + 2.59i)T - 7iT^{2} \)
11 \( 1 + (-2.14 - 5.18i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.984 + 0.407i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.979iT - 17T^{2} \)
19 \( 1 + (-5.68 - 2.35i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.70 + 3.70i)T + 23iT^{2} \)
29 \( 1 + (1.17 - 2.83i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.54T + 31T^{2} \)
37 \( 1 + (8.23 - 3.41i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.10 + 1.10i)T + 41iT^{2} \)
43 \( 1 + (3.47 + 8.37i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 3.15iT - 47T^{2} \)
53 \( 1 + (2.55 + 6.16i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (8.95 - 3.70i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.00 - 4.84i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.14 + 2.76i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-10.0 + 10.0i)T - 71iT^{2} \)
73 \( 1 + (-8.11 - 8.11i)T + 73iT^{2} \)
79 \( 1 - 0.155iT - 79T^{2} \)
83 \( 1 + (-5.13 - 2.12i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (6.15 - 6.15i)T - 89iT^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−13.74915183883831094815957995519, −12.29904549743330346247008458291, −11.81464330126326972088980136606, −10.59620361784684353175765940930, −9.575302208063653326883619667607, −8.038599149392844909064122943652, −7.08593280662099143083646012135, −4.93936250850858838668425683647, −3.73626550966903623812880685815, −1.69124892458864588971863194933, 3.53321672967151914680692181733, 4.96327178074623808121589857975, 5.99821013393399217255036106485, 7.81907645625966297181624084817, 8.554058839708713355074767975135, 9.480921772906754350166894306572, 11.37916014693078867616251399372, 12.05479650936063913162227992349, 13.73428606751168157295233410415, 14.31881798466901871342750644139

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