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Properties

Label 2-96-96.35-c1-0-9
Degree $2$
Conductor $96$
Sign $-0.290 + 0.956i$
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  + (−1.12 − 0.858i)2-s + (−0.0380 − 1.73i)3-s + (0.525 + 1.92i)4-s + (2.18 − 0.906i)5-s + (−1.44 + 1.97i)6-s + (−1.93 − 1.93i)7-s + (1.06 − 2.61i)8-s + (−2.99 + 0.131i)9-s + (−3.23 − 0.860i)10-s + (−1.42 + 0.590i)11-s + (3.32 − 0.983i)12-s + (−0.110 + 0.266i)13-s + (0.512 + 3.83i)14-s + (−1.65 − 3.75i)15-s + (−3.44 + 2.02i)16-s + 6.17·17-s + ⋯
L(s)  = 1  + (−0.794 − 0.607i)2-s + (−0.0219 − 0.999i)3-s + (0.262 + 0.964i)4-s + (0.978 − 0.405i)5-s + (−0.589 + 0.807i)6-s + (−0.730 − 0.730i)7-s + (0.376 − 0.926i)8-s + (−0.999 + 0.0439i)9-s + (−1.02 − 0.271i)10-s + (−0.429 + 0.177i)11-s + (0.958 − 0.283i)12-s + (−0.0306 + 0.0739i)13-s + (0.136 + 1.02i)14-s + (−0.426 − 0.969i)15-s + (−0.861 + 0.507i)16-s + 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 96,\ (\ :1/2),\ -0.290 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.439344 - 0.592571i\)
\(L(\frac12)\) \(\approx\) \(0.439344 - 0.592571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.858i)T \)
3 \( 1 + (0.0380 + 1.73i)T \)
good5 \( 1 + (-2.18 + 0.906i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.93 + 1.93i)T + 7iT^{2} \)
11 \( 1 + (1.42 - 0.590i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.110 - 0.266i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 6.17T + 17T^{2} \)
19 \( 1 + (-7.34 - 3.04i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.85 + 1.85i)T + 23iT^{2} \)
29 \( 1 + (2.11 - 5.10i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 3.42iT - 31T^{2} \)
37 \( 1 + (-2.52 - 6.09i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.753 + 0.753i)T - 41iT^{2} \)
43 \( 1 + (-1.57 - 3.79i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.54iT - 47T^{2} \)
53 \( 1 + (5.12 + 12.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.08 - 7.45i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.28 + 1.77i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (0.531 - 1.28i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (8.72 - 8.72i)T - 71iT^{2} \)
73 \( 1 + (2.73 + 2.73i)T + 73iT^{2} \)
79 \( 1 + 2.76T + 79T^{2} \)
83 \( 1 + (2.53 - 6.11i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.14 + 4.14i)T + 89iT^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−13.32685241606029866276209215161, −12.64133143754573266106633406770, −11.64194518635364042911981837841, −10.14760803341320852881312520642, −9.543854368073788689801486756333, −8.048482126646017623341020142355, −7.13133237678251076579935301309, −5.71021040262295793870726423824, −3.15527101616893724072605854783, −1.34654085515539269292616708823, 2.86751593819372063870867209762, 5.42478785316970525880587563539, 5.99050565308677116446023467747, 7.69092290335725325641668065304, 9.244835996233290433726924921058, 9.699936557113505054590762618536, 10.57913220906673614365058182470, 11.87664999949185858125339767021, 13.70523602342842152609396092513, 14.46682659224992548608773596899

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