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Properties

Label 2-968-11.3-c1-0-3
Degree $2$
Conductor $968$
Sign $-0.995 - 0.0913i$
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  + (0.791 + 2.43i)3-s + (0.454 + 0.330i)5-s + (−1.58 + 4.87i)7-s + (−2.88 + 2.09i)9-s + (−2.52 + 1.83i)13-s + (−0.444 + 1.36i)15-s + (−1.61 − 1.17i)17-s + (−1.23 − 3.80i)19-s − 13.1·21-s + 6.56·23-s + (−1.44 − 4.45i)25-s + (−1.16 − 0.845i)27-s + (0.965 − 2.97i)29-s + (1.16 − 0.845i)31-s + (−2.32 + 1.69i)35-s + ⋯
L(s)  = 1  + (0.457 + 1.40i)3-s + (0.203 + 0.147i)5-s + (−0.598 + 1.84i)7-s + (−0.960 + 0.697i)9-s + (−0.700 + 0.509i)13-s + (−0.114 + 0.353i)15-s + (−0.392 − 0.285i)17-s + (−0.283 − 0.872i)19-s − 2.86·21-s + 1.36·23-s + (−0.289 − 0.891i)25-s + (−0.223 − 0.162i)27-s + (0.179 − 0.551i)29-s + (0.209 − 0.151i)31-s + (−0.393 + 0.285i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.995 - 0.0913i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.995 - 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0658170 + 1.43731i\)
\(L(\frac12)\) \(\approx\) \(0.0658170 + 1.43731i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.791 - 2.43i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.454 - 0.330i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.58 - 4.87i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.52 - 1.83i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.61 + 1.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
29 \( 1 + (-0.965 + 2.97i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.16 + 0.845i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.06 - 3.27i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.20 - 6.77i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + (-2.47 - 7.60i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.43 + 2.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.95 - 12.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.76 - 4.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.43T + 67T^{2} \)
71 \( 1 + (2.98 + 2.16i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.965 - 2.97i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.32 + 1.69i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.38 + 5.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.68T + 89T^{2} \)
97 \( 1 + (9.25 - 6.72i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.12388986571256275520330525262, −9.460783479990128840940345639677, −9.038423021220904682254738746720, −8.368980515722994205453475904285, −6.90289240095816726126804083918, −5.98834629660488615547599272289, −5.02911929463829169735770419022, −4.34471638160178654180794641615, −2.87618707909981748798648320336, −2.53847686890663629065377261524, 0.61833655461894692779965450623, 1.72180329477055474111526302668, 3.06506352952664786982056685297, 4.03782266709592408977187728519, 5.37501816530791657306081412076, 6.62864613922746453110659849285, 7.11263749826913726339197269878, 7.68100998658943209303194557129, 8.580737090575792104926451815571, 9.630328424169517990415363963144

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