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Properties

Label 2-968-88.83-c1-0-24
Degree $2$
Conductor $968$
Sign $0.835 - 0.550i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (−1.34 + 0.437i)2-s + (−1.61 − 1.17i)3-s + (1.61 − 1.17i)4-s + (2.68 + 0.874i)6-s + (−1.66 + 2.28i)8-s + (0.309 + 0.951i)9-s − 4.00·12-s + (1.23 − 3.80i)16-s + (5.37 + 1.74i)17-s + (−0.831 − 1.14i)18-s + (−4.98 + 6.86i)19-s + (5.37 − 1.74i)24-s + (−4.04 − 2.93i)25-s + (−1.23 + 3.80i)27-s + 5.65i·32-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (−0.934 − 0.678i)3-s + (0.809 − 0.587i)4-s + (1.09 + 0.356i)6-s + (−0.587 + 0.809i)8-s + (0.103 + 0.317i)9-s − 1.15·12-s + (0.309 − 0.951i)16-s + (1.30 + 0.423i)17-s + (−0.195 − 0.269i)18-s + (−1.14 + 1.57i)19-s + (1.09 − 0.356i)24-s + (−0.809 − 0.587i)25-s + (−0.237 + 0.732i)27-s + 1.00i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.835 - 0.550i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.835 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558853 + 0.167547i\)
\(L(\frac12)\) \(\approx\) \(0.558853 + 0.167547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.437i)T \)
11 \( 1 \)
good3 \( 1 + (1.61 + 1.17i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-5.37 - 1.74i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.98 - 6.86i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.65 + 9.15i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.85 + 3.52i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.97 - 13.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.68 - 0.874i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (-3.09 - 9.51i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.14783039166987195237748556233, −9.312060517358157770901280512670, −8.136271010847294120084961356956, −7.73684117098528679620146327441, −6.60727792709843856422365277118, −6.04623483427527995404884435160, −5.39086487366831230193474437802, −3.76201506143394633877014224520, −2.10136897369292577540135847343, −0.963744034557341516210586540152, 0.53872596544596216882282243563, 2.25100305837123839125294460095, 3.52953968905620306409025625062, 4.69021359022113646716674632380, 5.66642622263868561760256356544, 6.58567607591960980926320047416, 7.51511792174876212162279854014, 8.396513193935239304897983819309, 9.377349573763453533830965900344, 9.959672915433369571630213149785

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