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Properties

Label 2-968-11.9-c1-0-13
Degree $2$
Conductor $968$
Sign $0.998 + 0.0475i$
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  + (−2.46 + 1.79i)3-s + (0.550 + 1.69i)5-s + (0.285 + 0.207i)7-s + (1.94 − 5.97i)9-s + (1.71 − 5.27i)13-s + (−4.39 − 3.19i)15-s + (−1.17 − 3.62i)17-s + (3.73 − 2.71i)19-s − 1.07·21-s − 7.00·23-s + (1.47 − 1.07i)25-s + (3.09 + 9.51i)27-s + (1.71 + 1.24i)29-s + (1.06 − 3.28i)31-s + (−0.194 + 0.598i)35-s + ⋯
L(s)  = 1  + (−1.42 + 1.03i)3-s + (0.246 + 0.757i)5-s + (0.107 + 0.0784i)7-s + (0.647 − 1.99i)9-s + (0.475 − 1.46i)13-s + (−1.13 − 0.823i)15-s + (−0.285 − 0.879i)17-s + (0.857 − 0.622i)19-s − 0.234·21-s − 1.46·23-s + (0.295 − 0.214i)25-s + (0.594 + 1.83i)27-s + (0.318 + 0.231i)29-s + (0.191 − 0.590i)31-s + (−0.0328 + 0.101i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.998 + 0.0475i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882968 - 0.0209955i\)
\(L(\frac12)\) \(\approx\) \(0.882968 - 0.0209955i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 - 1.79i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.550 - 1.69i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.285 - 0.207i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.71 + 5.27i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.17 + 3.62i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.73 + 2.71i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.00T + 23T^{2} \)
29 \( 1 + (-1.71 - 1.24i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.06 + 3.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.40 - 1.02i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.19 - 0.871i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
47 \( 1 + (-0.979 + 0.711i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.44 + 7.53i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.5 - 1.81i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.35 - 7.23i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + (0.732 + 2.25i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.15 - 2.29i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.26 - 3.90i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.252 + 0.778i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.92T + 89T^{2} \)
97 \( 1 + (2.92 - 9.01i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.12222154262707644162646495784, −9.651647004952648288474445604591, −8.421238787511529237991543705845, −7.24689306722761477518479603960, −6.34244104192234779609924030441, −5.62323292815801538061604409078, −4.94452658516651736838170022178, −3.84880899784617972843002296594, −2.77525223477797365704648769438, −0.58740431846304001147640126214, 1.15630007055096233078225971207, 1.93022351177459967814413950346, 4.05016070354977941547841500529, 4.96120802478850888578047319823, 5.91890437587490165666538944035, 6.39233459214346242393676821899, 7.35734468242280426309386159918, 8.207423467258822352735577117969, 9.166561763788102882464552356125, 10.19670250919378741249032341873

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