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Properties

Label 2-968-11.5-c1-0-20
Degree $2$
Conductor $968$
Sign $0.719 + 0.694i$
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  + (1.26 + 0.917i)3-s + (1.10 − 3.38i)5-s + (2.52 − 1.83i)7-s + (−0.173 − 0.534i)9-s + (1.58 + 4.87i)13-s + (4.49 − 3.26i)15-s + (−0.618 + 1.90i)17-s + (−3.23 − 2.35i)19-s + 4.87·21-s + 2.43·23-s + (−6.21 − 4.51i)25-s + (1.71 − 5.28i)27-s + (−4.14 + 3.01i)29-s + (−1.71 − 5.28i)31-s + (−3.43 − 10.5i)35-s + ⋯
L(s)  = 1  + (0.729 + 0.529i)3-s + (0.492 − 1.51i)5-s + (0.954 − 0.693i)7-s + (−0.0578 − 0.178i)9-s + (0.439 + 1.35i)13-s + (1.16 − 0.844i)15-s + (−0.149 + 0.461i)17-s + (−0.742 − 0.539i)19-s + 1.06·21-s + 0.508·23-s + (−1.24 − 0.903i)25-s + (0.330 − 1.01i)27-s + (−0.769 + 0.559i)29-s + (−0.308 − 0.949i)31-s + (−0.580 − 1.78i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.719 + 0.694i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.719 + 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21583 - 0.894536i\)
\(L(\frac12)\) \(\approx\) \(2.21583 - 0.894536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-1.26 - 0.917i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-1.10 + 3.38i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.52 + 1.83i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.58 - 4.87i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.618 - 1.90i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 + (4.14 - 3.01i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.71 + 5.28i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.11 + 4.44i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.908 + 0.660i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.12T + 43T^{2} \)
47 \( 1 + (6.47 + 4.70i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.78 - 11.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.31 - 4.58i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.347 - 1.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.56T + 67T^{2} \)
71 \( 1 + (2.68 - 8.25i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.14 + 3.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.43 - 10.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.270 - 0.833i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2.68T + 89T^{2} \)
97 \( 1 + (-4.80 - 14.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.500579213291431501660018563458, −9.102391111725972171281881122226, −8.536209358350879344563710811244, −7.66744474144696965228639145427, −6.44952071866094538531992382164, −5.36160871266765370650667401699, −4.28630494383086740624785629146, −4.07849986691324809966319676530, −2.21391887938293164863244266788, −1.12832429456611363563797337798, 1.80399854171188810184508823344, 2.62281650440425454031680776544, 3.37426247294990576563010306968, 5.03458978848351632435229361207, 5.92124025910157790633164457050, 6.80102026698598313802025402303, 7.81063212058076158875050098711, 8.172424360793216757317980087174, 9.198325485511215492934982689353, 10.24859738800171913311411668786

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