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Properties

Label 2-96-96.35-c1-0-0
Degree $2$
Conductor $96$
Sign $-0.959 - 0.282i$
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  + (−0.816 + 1.15i)2-s + (−0.266 + 1.71i)3-s + (−0.667 − 1.88i)4-s + (−3.14 + 1.30i)5-s + (−1.75 − 1.70i)6-s + (0.663 + 0.663i)7-s + (2.72 + 0.769i)8-s + (−2.85 − 0.911i)9-s + (1.06 − 4.69i)10-s + (1.91 − 0.794i)11-s + (3.40 − 0.639i)12-s + (−2.31 + 5.59i)13-s + (−1.30 + 0.224i)14-s + (−1.39 − 5.73i)15-s + (−3.11 + 2.51i)16-s + 2.24·17-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)2-s + (−0.153 + 0.988i)3-s + (−0.333 − 0.942i)4-s + (−1.40 + 0.582i)5-s + (−0.718 − 0.695i)6-s + (0.250 + 0.250i)7-s + (0.962 + 0.271i)8-s + (−0.952 − 0.303i)9-s + (0.336 − 1.48i)10-s + (0.578 − 0.239i)11-s + (0.982 − 0.184i)12-s + (−0.642 + 1.55i)13-s + (−0.349 + 0.0600i)14-s + (−0.359 − 1.48i)15-s + (−0.777 + 0.628i)16-s + 0.545·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $-0.959 - 0.282i$
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: \((2,\ 96,\ (\ :1/2),\ -0.959 - 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0744512 + 0.516966i\)
\(L(\frac12)\) \(\approx\) \(0.0744512 + 0.516966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.816 - 1.15i)T \)
3 \( 1 + (0.266 - 1.71i)T \)
good5 \( 1 + (3.14 - 1.30i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.663 - 0.663i)T + 7iT^{2} \)
11 \( 1 + (-1.91 + 0.794i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (2.31 - 5.59i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 2.24T + 17T^{2} \)
19 \( 1 + (-3.08 - 1.27i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.32 - 4.32i)T + 23iT^{2} \)
29 \( 1 + (-0.546 + 1.32i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 2.34iT - 31T^{2} \)
37 \( 1 + (0.324 + 0.783i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.73 - 4.73i)T - 41iT^{2} \)
43 \( 1 + (-0.951 - 2.29i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.02iT - 47T^{2} \)
53 \( 1 + (3.49 + 8.43i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.11 - 7.51i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.01 + 0.421i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.46 + 8.35i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-0.167 + 0.167i)T - 71iT^{2} \)
73 \( 1 + (-3.86 - 3.86i)T + 73iT^{2} \)
79 \( 1 - 2.44T + 79T^{2} \)
83 \( 1 + (5.17 - 12.4i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-6.63 - 6.63i)T + 89iT^{2} \)
97 \( 1 - 4.48T + 97T^{2} \)
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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

−14.79383239491202038862408538595, −14.07079050237934045299288050659, −11.69520218073526599156710751889, −11.35748486829354417757664028201, −9.917333098403515350266167446285, −8.979528572988292838176364144610, −7.79684356706694414748841746577, −6.67206076769262253623383296250, −5.04312511814080004215830366760, −3.77874405060612508662779935532, 0.816397880858005899972243157475, 3.18193012304107588745631421887, 4.89202116122090971120232980192, 7.25607126577126439022889616882, 7.900703243787326785886293869697, 8.863368620891103634414863639078, 10.52844616922055271860979492797, 11.58781269780140765292489688775, 12.33801615669726817460899396765, 12.89216012912239142860208061806

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