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Properties

Label 2-968-88.35-c1-0-27
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: $\chi_{968} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.835 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496317 + 1.65545i\)
\(L(\frac12)\) \(\approx\) \(0.496317 + 1.65545i\)
\(L(\frac{3}{2})\) 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

−10.49434158029550317485733808663, −9.840192829061610146320015848972, −8.540450257088924379243311148950, −7.65348788626745708603684474279, −6.66619814899980988749294469482, −5.78454532203826157507132540632, −5.25305131733330759763756346926, −4.26981576028405297166661535629, −3.50605190035201964967214033650, −1.96886419976735374995847307922, 0.63562563617863124686308639412, 2.07369768795048370762373777624, 3.25028559441328807126005541432, 4.58361928570564281867236966916, 5.24994644623853423540227231562, 6.25387086412479207592475097400, 6.80883063361230721768532010248, 7.55483968087288862841259738207, 8.979592581770203108916053143035, 9.895355947582561651143420157025

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