Properties

Label 2-968-11.3-c1-0-26
Degree $2$
Conductor $968$
Sign $-0.220 - 0.975i$
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 − 3.07i)3-s + (−1.80 − 1.31i)5-s + (1 − 3.07i)7-s + (−6.04 + 4.39i)9-s + (−1.42 + 1.03i)13-s + (−2.23 + 6.88i)15-s + (−0.809 − 0.587i)17-s + (−1.76 − 5.42i)19-s − 10.4·21-s + 0.763·23-s + (11.7 + 8.50i)27-s + (0.545 − 1.67i)29-s + (−3.85 + 2.80i)31-s + (−5.85 + 4.25i)35-s + (−0.0729 + 0.224i)37-s + ⋯
L(s)  = 1  + (−0.577 − 1.77i)3-s + (−0.809 − 0.587i)5-s + (0.377 − 1.16i)7-s + (−2.01 + 1.46i)9-s + (−0.395 + 0.287i)13-s + (−0.577 + 1.77i)15-s + (−0.196 − 0.142i)17-s + (−0.404 − 1.24i)19-s − 2.28·21-s + 0.159·23-s + (2.25 + 1.63i)27-s + (0.101 − 0.311i)29-s + (−0.692 + 0.502i)31-s + (−0.989 + 0.718i)35-s + (−0.0119 + 0.0369i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.220 - 0.975i$
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: \((2,\ 968,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328893 + 0.411675i\)
\(L(\frac12)\) \(\approx\) \(0.328893 + 0.411675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 3.07i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.80 + 1.31i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1 + 3.07i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.42 - 1.03i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.76 + 5.42i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.763T + 23T^{2} \)
29 \( 1 + (-0.545 + 1.67i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.85 - 2.80i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0729 - 0.224i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.30 - 7.10i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (-1.76 - 5.42i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.6 + 7.74i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.70 - 5.25i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (12.0 + 8.78i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.763T + 67T^{2} \)
71 \( 1 + (-3.23 - 2.35i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.09 + 3.35i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.85 + 4.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.85 + 7.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (9.66 - 7.02i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.160262980771609527908596264313, −8.241087410729814498746274962250, −7.54132010550046829909678069797, −7.09190624454322674520801151687, −6.24167042191870860365925790908, −5.02399253648605318678868001626, −4.22579951780246722986993915786, −2.56491013086633090863447980835, −1.23048101187121523318580826462, −0.28902981943076688392537877776, 2.58041096389708111426869710849, 3.68390527296291634452348740583, 4.32958950669168270750452223938, 5.50525669024145918136587881499, 5.84303951296651194035829911721, 7.28238723901988899318204966459, 8.391190982730801812308613788075, 9.057561921907013914504856261222, 9.868111631685644181615284277282, 10.77730719715066765347541875032

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