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Properties

Label 2-96-96.11-c1-0-10
Degree $2$
Conductor $96$
Sign $0.901 + 0.431i$
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 + (1.25 + 1.19i)3-s + (0.525 − 1.92i)4-s + (−2.18 − 0.906i)5-s + (2.43 + 0.271i)6-s + (−1.93 + 1.93i)7-s + (−1.06 − 2.61i)8-s + (0.131 + 2.99i)9-s + (−3.23 + 0.860i)10-s + (1.42 + 0.590i)11-s + (2.96 − 1.78i)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 + (0.722 + 0.691i)3-s + (0.262 − 0.964i)4-s + (−0.978 − 0.405i)5-s + (0.993 + 0.110i)6-s + (−0.730 + 0.730i)7-s + (−0.376 − 0.926i)8-s + (0.0439 + 0.999i)9-s + (−1.02 + 0.271i)10-s + (0.429 + 0.177i)11-s + (0.857 − 0.515i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.901 + 0.431i$
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: \((2,\ 96,\ (\ :1/2),\ 0.901 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45185 - 0.329737i\)
\(L(\frac12)\) \(\approx\) \(1.45185 - 0.329737i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 1.19i)T \)
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

−13.85110013828683869705879524832, −12.83551059772402366871459232238, −11.86987700475859673609345240395, −10.87891820036591054976898971344, −9.504550967288414459806391294757, −8.772762256668667408472201493590, −6.99709329825917986532409392182, −5.17907538370279207388219703655, −4.02940144255409099178863005648, −2.79576473162976872116598137351, 3.12697474783330842216541889276, 4.11139208652276285962765662454, 6.32961762606015535000570182470, 7.23365033035994521538045785482, 8.002413509863123271143330400601, 9.413747158280631815671417151205, 11.33289399771583409966813133554, 12.13947290712595483371074392106, 13.40089748703172323572079138781, 13.82647532902952128440864115770

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